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•

§-■

A COURSE IN
EXTERIOR BALLISTICS

ORDNANCE TEXTBOOK

PREPARED BY THE

ORDNANCE DEPARTMENT

December, 1920

UJa-..,'..; : '- ^ %.\.

FEB 13 1922

-f ^.

< *<'

./'

r J

.^T/ /

War Defabtmbnt.

Document No. 1051.

Office of The Adjutant General,

(The material for this textbook is taken from a course of lectures on Ballistic Mathe-
matics and Ballistics given at the Ordnance School of Application during the year
1919-20.)

caaBSB a xxukbior ejlllisiigs

nil
19

P.
p.

5
U

p.
p.
p.

30
39
39

P. 39

P« 44

P.
p.

BBBATi

line 4 from bottom; I should be 1.
line 11 from bottom of text; before
**or see,** Insert footnote reference, 2.
bottom; add footnote 2:*See
supplement F***

line 9; Integral should haye loiier
limit, tn.

prob. 25; xg should be x'^.
line 11 from bottom; delete <*the.**
line 5 from bottom; delete **for
eaoh arc***

line 4 from bottom; delete ** differ-
ence for each arc, and hence the
correct range***

line 20; after ** theoretic ally,** add
footeote reference, 3.
bottom; add footnote 3:**See p« 63/
footnote, last eq.; Parenthesis
should read : (y| 4. ^H?) .
line 13; for **a funoXion of,** read
'*dependent on***

line 17; for **equation8 47,** read
"equation 47."

line 7 from bottom; for **boo]c,<*
read **chapter«**

line 6 from bottom; for '*some in
Chapter XI, " read '*that for site.**
line 5 from bottom; for **A^** read

**<r^ .** ^

line 19; instead of **whence,** the
line should read: *'whenoe, bgr
theorem 9, page 21 1**
line 9 from bottom; add foolaote
reference, 2«

line 4 fl*om bottom; add footnote
reference, 3«

bottom: add footnotes :"S 3ee p* 44,**
and **3 See Problem 55, p* 69, post.*"
line 3; first word should be '*in.**

fc— Bit

P. 67

P. 69

P* 70

P* 71

P. 76
P. 81

P. 81

P. 84

P. 85

P. 88
P. 88

P. 88

P. 93

P. 95

P. 97

P. 113

P.114
P.117
P. 126

last line; to read: **radias of the

earth, andcj was the angle of fall.**

lines 1, 2; change "x - X" to "X -

X** in eaoh line* ^

lines 12, 13; change "As" to " d^a'»

in each line* '

fig. 6; the "J" which marks the angle

should be **j."*

first eq«; "(S f J)" should be

eq» 79; delete the comma in the 1019.

eq« 94; (x*p - y'V )" should be

«(-i7y» -fx').*

eq» 95; change '*ooso(" to "*sinoc" in

first eq« only*

85; column to right of the T column

should have had all horisontal lines

broken, to iniicate that several

columns have been omitted*

jfo« 56; Integral should have upper

limit, "T."

first eq.; "^ Mt ^ should be "/^to*"

values of A, B, G; lower limits on

integrals are 0.

value of A; "(x'p - y'l^ )" should be

n(Vy» -pxM.''

line 7 from bottom; integral should
have upper limit, "T"
fig* 13; reference should be to fig*
12, instead of to fig* 1.
footnote; the word **second" should
not be italicized*
line 3 from bottom of text; delete
the footnote reference*
line 1; line should read: "Substitut-
ing from equations 117 and 118 in
equations 116)**

last 5 lines; change "dx" to ''dX,**
four times*

footnote; the first ^ in the equation
should have for subscript, "t • l.**
col, 2; "tA " should be "t^ "

k •»

mat
191€

WAK DEPARTMENT,
Washington, December 14, 1920.

The following publication, entitled ''A Course in Exterior Ballis-
tics/' is prepared for the information and guidance of all concerned,
[062.1, A. G. O.]

By order of the Secretary of War:

PEYTON C. MARCH,

Major Oeneralf Chief of Staff.
Official:

P. C. HARRIS,

The Adjutant General.

CONTENTS.

Page.

Introduction 7

Ghaftbr I. Partial differentiation 10

II. Successive approximations 14

III. Effect of differential variations 18

IV. Finite differences 28

V. Elements of the trajectory 34

VI. History of exterior ballistics 38

VII. The ifiLOtion of a projectile 43

VIII. Computation of trajectories 47

IX. Derivation of auxiliary variables 53

X. Range correction formulas 59

XI. Angle of departure correction formulas 70

XII. Deflection formulas 73

XIII. Rotation of the earth 76

XIV. Computation of differential corrections 83

XV. Weighting factors 91

XVI. Construction of a range table 96

Supplement A. Trajectory computation by the tangent reciprocal method... 103

B. Explanation of the signs in the computation of differential

corrections 106

C. Dimensions of ballistic symbols 107

D. Antiaircraft fire 109

E. Derivation of two equations of Chapter VII. Ill

F. A derivation of Theorem 1 114

G. New methods of trajectory computation 116

H. Note on advancing difference formulas 120

Index , 123

EXTERIOR BALLISTICS.

INTRODUCTION.

The work of the ballistic computer is divided into three parts:
(1) the computation of the elements of standard trajectories; (2) the
computation of differential corrections, whereby the elements of a
standard trajectory may be corrected for nonstandard conditions;
and (3) the utilization of the foregoing to construct range tables
from firing records.

As is shown in C3iapter VI, the World War ushered in a new era
in the handling of ballistic problems. This may be called the period
of numerical integration.

The approximations, which had been used to modify the simple
Newtonian equations of motion into such form that they could be
fomaUy integrated, gave place to precise numerical inteVatiou of
these equations in their original form.

Practically parallel prosress was made in all of the allied countries
duringthewar.

In America the great step in the computation of trajectories was
the introduction of nimierical integration. Numerical integration
had long been used in astronomical calculations, and so it was natural
that an astronomer and mathematician, Prof. F. R. Moulton, of the
University of Chicago, while serving as a major in the Ordnance Depart-
ment of the United States Army, should have applied this method to
ballistics. The practical work of this method was materially reduced
by formulas for '^integrating ahead" later introduced.

The work can be still further reduced by a variant of Maj. Moulton's
method, known as the tangent-reciprocal method. But this method
has the pedagogical drawback of obscuring the physical meaning of
the steps involved, and hence will not be given first place in this
book.*

Another improvement is the change in the analytic interpretation
of the equations of motion. Formerly the x-axis was conceived of
as tangent to the earth at the gun, the system being Cartesian. In
the modem conception, the « of a point is measured along the curved
surface of the earth, and the y is measured vertically from this surface.

The progress in the computation of the differential corrections has
involved more steps. Differential equations for the corrections were
devised at the same time that numerical integration was introduced.

1 See Supplement A.

8 COURSE IN EXTERIOR BALLISTICS.

But these had the difficulty of requiring an independent computa*
tion for each correction.

This difficulty was removed by the discovery at Aberdeen of a
method of solution by means of a set of adjoint equations involving
several auxiliary variables. All the differential corrections can be
expressed in terms of these auxiliary variables. A physical deriva*
tion of these variables, and of the corrections based upon them, was
later found. Matters were further simplified by reducing all of these
variables to expressions in terms of one variable and its derivatives*
A physical derivation of this step was at once forthcoming.

One further step should be noted, namely the development of the
weighting-factor curves for zero elevation, which have been of great
value in interpolation.

The present method of constructing range tables out of firing
records is a logical result of substituting for the IngaJls tables the
new methods of computation. Tables, to take the place of the com-
putations now necessary, are now being constructed by the tech-
nical staff at Washington.

The credit for the above-described development is largely due to
Mr. J. J. Amaud, Master Computer, Ordnance Department; Prof.
A. A. Bennett, of the University of Texas, then Captain, Ordnance
Department, U. S. Army; Prof. G. A. Bliss, of the University of
Chicago, Technical Expert, Aberdeen; Mr. Philip Franklin, Com-
puter, Aberdeen; Dr. T. H. Gronwall, Mathematics and Dynamics
Expert, Technical Staff; Prof. H. H. Mitchell, of the University of
Pennsylvania, Master Computer, who organized the range table
computation work at Aberdeen; Dr. J. F. Ritt, of Columbia Uni-
versity, Master Computer, Technical Staff, and their associates, in
addition to those mentioned elsewhere herein.

Most of the written material on the ballistic progress made during
the war consists of scattered blue prints, some printed at Aberdeen and
some at Washington. These pamphlets overlap in spots, contain
some hiatuses, and do not agree in symbology and nomenclature.
Through a three-cornered correspondence between the Technical Staff,
Ordnance Office, (War Department), at Washington, D. C, and the
Ordnance School of Application and the Ballistic Section, Aberdeen
Proving Ground, Md., a uniform symbology and nomenclature have
been established as standard.

The first course of instruction in these new ballistic methods ever
given in this country was given at the Ordnance School of Applica-
tion in the winter of 1919-20 by Capt. Roger Sherman Hoar, Coast
Artillery, then in charge of the Ballistic Section of the Proof Depart-
ment at Aberdeen. This present book is based upon the papers
used in that course, and uses the standard symbology and nomen-
clature established as above.

INTRODUCTION. 9

It is assumed that the student is thoroughly grounded in algebra
and plane trigonometry, and knows enough calculus to appreciate
the meaning of a derivative, a differential, and a definite integral.
On that basis, this book gives, in Chapters I to IV, the irreducible
minimum of higher mathematics necessary to understand all points
involved in the later chapters.

The book then takes up in succession : An introduction to modern
ballistic methods (Chaps. V and VI); the computation of trajecto-
ries (Chaps. VTI and VIII); the computation of differential correc-
tions (Chaps. IX to XV) ; and the construction of range tables (Chap.
XVI). Alternative methods, elaborations of certain points, and a
brief mention of the more involved mathematical processes necessary
to the computation of antiaircraft range tables are reserved for
supplements.

Each chapter is followed by a series of questions, designed to bring
out the salient features of the chapter. The answers to most of
these questions will be found categorically stated in the text, but
some under each chapter will require a small degree of original
thought on the part of the student.

Throughout the book the attempt is made to explain as much as
possible from the viewpoint of physics rather than from the view-
point of abstract mathematics.

CHAPTER I.
PARTIAL DIFFERENTIATION.

Before defining the ''partial differentiation" of a function, let us
define the word *' function." u is called a ''function" of x, j/, z, etc«
if, when a;, y, s, etc., are given, the value of t^ is determined. Note
the broadness of this definition. Thus u = xy can be regarded as a
function of x, y, and s, although it is evident that u is not in the
least dependent on z.

The functional relation is expressed in the general form:

"^^/Ca;, y, 2, . . . ).

One should be siu'e to notice the fundamental fact that this ex-
pression, as it stands, does not take into consideration any relation-
ship which may exist between any of the variables in the parenthesis;
in other words, these variables may or may not be independent in
that expression.

Usually the fimctional equation can be altered so as to make x
explicitly a function of u, y, z, etc. ; y expHcitly a function of u, x, z,
etc. In case such a conversion is either impossible or even merely
inconvenient, it is better to regard the equation, in its original form,
as defining u as a function of x, y, z, etc., or x as a fimction of u, y, 2,
etc.; and to differentiate it as it stands, using the differential method.
Such an equation is that in problem 3, to follow. Thus, in an equa-
tion containing n variables, any one of these variables can generally
be regarded as dependent, and the remaining n-1 as independent.

We are now in a position to define the term ''partial derivative."
If u=/ {x, y, z . . . ), then the partial derivative of u with
respect to x is obtained by treating as constants all the other variables
in the parenthesis, and differentia'ting with respect to x, in the ordi-
nary manner.

^iji du

But the partial derivative is written >r-, instead of -^, so as to in-

indicate that x is one of several independent variables, instead of being

the sole independent variable. Thus ^ means the rate of change

of u with respect to a change in x alone out of several independent
variables.
An important point to note in this connection is that the symbol

J— is highly ambiguous, i. e., it has entirely different meanings, and its

partial derivatives have entirely different values, according to what
10

I. PARTIAL. DIFFEBBNTIATiON. 11

are regarded as the independent variables. To avoid this ambiguity,
the subscript notation should be used. Thus -j~^ means that u
should be expressed as a function of Zj y, and z, and then differentiated
with respect to x: —^ — means that u should be expressed as a func-
tion of X and Y before differentiating, etc.*

In practice, the subscripts may be omitted whenever it is self-
evident, from the problem, just what are to be treated as the inde-
pendent variables in each differentiation.

PROBLEMS.

Find ^9 ^ and ^ in each of the following:

(1) 2 = a: log y.

(2) z^ax^-hZbxy-of/.

(4) x^-hy^ + z'^a^

All of the theorems of partial differentiation can be derived as
special cases of the following general theorem:

1. If u is a function of X, Y, Z - - > , each of these in turn
being a function of x, y, z - • • , then {with certain assumptions as to
continuity):

dx ^ dX ' dx ^ dY ' dx '^'"

and similarly for ^'' * ' , etc.

This theorem can be derived by the use of imdetermined coeflGl-
cients, or see Osgood, p. 296.

This theorem is called '^ general," not in the sense that it is the
fundamental basis of the other theorems, for many of them have a
simpler derivation, and some, in fact, may serve as' steps in the deri-
vation of theorem 1. Nor is it advisable to use theorem 1 when some
simpler formula is available. But, as theorem 1 has a form easy to
remember, and as any other formula can be derived as a special case
thereof, it serves as a good memory peg on which to hang the whole
subject of partial differentiation.

The student should particularly note at this juncture that, although
any partial derivative may be obtained by the differential method;

1 For further expositiOD of the subscript notation, see Osgood, Differential and Integral Odculus, p. 306.

12 COUBSB IK BXTEBIOR BAUJSTIGS.

yet, once formed, its numerator and denominator are inseparable and
neither can be canceled. Witness the absurd results which would
follow from performing all possible cancellations in theorem 1.

The following general principles may be used in deriving special
formulas from theorem 1 :

(a) Whenever in partial differentiation any given variable is re-
garded as dependent on one independent variable alone, then, in the
expression for the derivative of the former with respect to the latter,,
the operator d should be changed to the operator d.

(6) The derivative of a variable, with respect to another variable
of which the first is regarded as independent, is zero.

(<i) If, for all values of its variables, a given function is explicitly
a constant, then the derivative of that function with respect to any
of such variables is zero.

The following special formulas may be derived from theorem 1 :

This is called the expression for the ''total differential" of u.

3« If u he a particular function of X and Yj namely X Y, then:

du^^dX dY
bx '^ dx dx '

which is the analog of the following formula of total differentiation :

du^YdX + XdY.

XY
4. Ifu egudla y > then:

iyg YbX XbY u^Z ubX ubY u dZ
dx"Z dx"^Z dx Zdx^X dx'Ydx Z dx '

6. Ifu equals Yx, then:

— = x—+Y
dx dx

Similar special formulas can be derived for other special relations:
which may exist between u, Xy Y, Z, etc.

PROBLEMS.

Note — The following problems should be treated exactly as though the symbolcr
2^, 7f^^ Ey etc., were the simpler-looking symbols of the preceding problems and
explanation. For the purposes of any given differentiation, x, zf, oif'y y, j^, y^\

T. PABTIAL DIFFEEESNTIATION. 13

*nd E have no relation to each other except that given in the hypotheses. But the
work on these problems should be carefully saved for later reference. In Chapter V
a meaning will be given to each of these symbols, and in Chapter IX it will be seen
that these problems, taken in order, constitute almost the entire proof of some im*
portant ballistic formulas.

PROBLEMS.

(5) Given that x" and y" are each functions of the independent
Tariables x', y', and y, evaluate dec" and dy" in terms of dx\ dy',
and dy and some partial derivatives.

For problems 6 to 13, the following is given:

x'' = -Ex'

y'^^^Ey'^g

where gr is a constant; and ^is a '' function" of x', y', y and Xy although
not dependent on ar.

The foregoing problems can be done either by one of the special
theorems or by theorem 1. It is advisable not to use theorem 1;
but if it be used, care should be taken to observe that x' and y'
each enter into the expressions for x" and y" in two capacities:
i, e., as a variable of the X sort and as a variable of the x sort. To
illustrate this, perform the following problem:

(14) x^'^EZ

E =/ (x', y\ y)
Z =x'

Evaluate ^-7- by theorem 1. Also by theorem 5.

Questions on Chapter I.

1. What is a partial derivative?

2. How, in general, is partial differentiation performed?

3. Does XT ' 5^ equal ^ ? If so, why? If not, why not?

4. If w=f (p, qj r), and ■>-- is evaluated in terms of p, q^ and r, and we then learn
^hat a certain special relationship— extraneous to the equation «?=/ (p, g, r)— -exista
between p, g, and r, will this fact change the value of ^? If the special relation-

iship is such that w equals a constant, will that fact make >g-=iO? If so, why? If

jiot, why not?

6. Explain the meaning of, and the need for, the subscript notation.

CHAPTER II.

SUCCESSIVE APPROXIMATIONS.

An equation which determines the. numerical value of a quantity
may generally be expressed in a variety of alternative ways. Thus
the fact that x is the square-root of 'W may be expressed:

(a) x— ± -v^
(J) X---0

x^wy
1

^ X

Here (a) defines x explicitly; (b) defines x implicitly; and (c) consists
of simultaneous equations in x and y, each defined as a function of
the other.

Forms analagous to (a) are not always forthcoming. Consider, for

example, the determination of a number x, such that x is equal to 500,

plus 1,000 times its own trigonometric sine, x being measured in

minutes of arc. There is no known explicit form for x. We have,

however :

(6) X = 500 + 1,000 sin X.

. V fx = 5C

x = 500+1/

000 sin X

(c,)

X = 500 y
y^l + iz
L2 = sin X

An implicit equation may frequently be replaced by a system of
equations (similar in general to equations of the sort c) so chosen as
to be convenient for solution by a computational procedure known as
"successive approximations.'^ This, as its name suggests, is a method
of starting with an apt number, largely arbitrary, and by successive
substitutions securing a sequence of approximate evaluations of x,
approaching the precise value to within any desired degree of precision

n. SUCCESSIVE APPROXIMATIONS. 16

For example, extract the square root of w by the following equa-
tions:

?/ =

a:-2 (y + ^^

Expressed as a formula:

-1

Extracting the square root of 2 by this pair of equations, first
assuming 2 to be approximately its own root, gives the following suc-
cessive values:

n X y

1 2 1

2 f I

3 ii ^

A 577

and so on. Expressed in ordinary language, we have the following
rule for the extraction of square root: Divide the number by an ap-
proximation to the square root desired; the arithmetic mean of the
divisor and quotient is a new approximation.

It can be shown that if any approximate square root checks with
the preceding approximate root to n figures, then the new approxi-
mate root is correct to at least 2n-l figures.

, Had a negative value been taken initially for x^, the negative
square root of 2 would have been approached. The separating value,
zero, causes the method to fail, as one would expect.

An attempt to extract the square root of 2 by the equations:

y-i

x^wy

lowi:

ng results :

COUBSE IN EXTERIOR BALLISTICS.

and so on. A similar repetition would occur for any initial value for
Xj other than zero or infinity. This shows that not all sets of simul-
taneous equations are adapted to solution by successive approxima-
tions.

The method of successive approximations has the advantage that
the accuracy of each step is independent of the accuracy of the pre-
ceding step. A mistake in any single step, therefore, while it may
prolong the work, will not vitiate the final result.

PROBLEMS.

(15) Extract the square root of 100, taking 25 as the approximate
square root and carrying each division to three decimal places. Con-
tinue imtil two successive answers check to three decimal places.

(16) Assume that 1.41459 is an approximation to the square root
of 2. Perform one step of getting the root with greater precision,
and give the result to only the number of places certain to be correct.

(17) Replace x = 500 + 1,000 sin x, by the system:

i/ = 2,000 sin a;

and use as formulas for successive approximation:

^n = 2,000 sin Xn

X is expressed in minutes of arc. Find x and y correct to the nearest
unit.

(18) y is a tabular function of the time, t. The following is a tabu-
lation showing the value of y corresponding to each of certain values

of e.

1, 569. 4

1, 052. 4

-517.

•

512.5

-539. 9

-22.9

-46.6

-559. 1

-19. 2

+3.7

Cj b, and c are, respectively, the first, second, and third differences of
y; i. e., the a of any line is obtained by subtracting, from the y of
that line, the y of the line before; the h of any fine is obtained by
subtracting, from the a of that line, the a of the line before, etc.

II. SUCCESSIVE APPROXIMATIONS. 17

To find the value of t corresponding to some given, non-tabular
value of y:
Let ]/t=the value of y at time, t;

yo = the value of y at the time, to]
and i» the tabular interval in U

Kepresenting by M the value of -^^, which is to be considered with
its algebraic sign, the formula to be used is:

6. A<- y^^

^l+M, , (1 +A0 (2 +A()

where a, 6, and e are the values taken /rom the same line on which to
and yo occur. The A^ so obtained must be multiplied by i, to obtain

"~ fro*

The formula is solved by successive approximation, the first
approximation being

Find the value of t corresponding to yt = 0, using to = 64. Check by
using to = 66.

This formula, obtainable from the result of problem 30, page 31,
may be used to interpolate. in either direction from any tabulated
values, but it requires the use of ^'receding differences;" i. e., differ-
ences that, in the method of writing used above, occur on the same
line with to and yo^

At the close of Chapter IV problems will be given involving a com-
bination of numerical integration and successive approximations.

Questions on Chapter II.

1. Define ' ' successive approximations. ' '

2. Give the rule for extraction of square root.

3. Is any set of simultaneous equations solvable by successive approximations?

4. What is the test of solvability?

5. What are the two chief advantages of this method?

1 When it is required to interpolate forward /rom the first item in a table, e. g., downward from the <-»60 in
the table above, the formula for "advancing differences'' must be used with the corresponding values of
y, a, b, and c; e. g., in this case, y= l,5e9.4; o— —517.0; 6— —22.9; and c=- +3.7. The "advancing difference"
formula may be obtained from by changing the sign of every At throughout the equation, and of the
odd-ordered differences, a and c. See Supplement H.

24647—21 2

CHAPTER III.

EFFECT OF DIFFERENTIAL YARUTIONS.

This chapter deals with the mathematical determination of the
effect of a disturbance, on the subsequent motion of a particle which
moves (except for the disturbance) according to some definite differ-
ential equations of motion.

Let us consider a particle moving in time, along a plane trajectorj •
' ' Trajectory' ' is here used in the general sense of the path of any moving
particle, rather than in the specialized ballistics sense of the path of
a projectile, or the still further specialized sense which will be the
meaning employed in later chapters, namely, the path of a projectile
moving under certain so-called standard conditions as to atmosphere,
wind, gravity, etc.

Reverting, then, to the motion of a particle along its trajectory,
it is evident that at any given instant of time (f) there will be a

uniquely corresponding value
of x, y, x', y', x", y", etc.,
where x and y are the coordi-
nates of the particle, x' and
y' the two components of
velocity, x" and y" the two
components of acceleration,
etc. Primes are thus seen
to represent time derivatives,
and will be used in that sense throughout this book. In this chapter
the general symbol u may be used in place of x, j/, x\ y\ x", and y",
in theorems true as to any of them. Hence u, also, is a function of t.
In the illustrative examples of this chapter, t^x coordinates and
t,y coordinates will frequently be employed, in order that the student
may become accustomed to considering the elements of the tra-
jectory as separately plotted against time, and to using time deriva^
tives.

In the case of motion of the sort which will be considered in this
book, and in fact in the case of most motions, the values of x", y",
and higher derivatives are, in the absence of disturbing causes, de-
termined, for any instant t, by the values of x, y, x' and y', or some
of them; and hence x'', y", etc., need not be discussed for the
present.

The path of the particle will, of course, be a single curve, which may
be graphed by plotting y against x. But a chronological record of its
18

in. EFFECT OF DIFFEBBNTIAL VABIATIONS. 19

motion may be represented more completely by ftAht curves, obtained
by plotting a5, y, x', and y', respectively, against U

Consider now any one of these four curves, represented by ABG
in figure 1.

Suppose a disturbance from B to £J, so that the curve takes the
shape BE during the interval of disturbance, and suppose that
thereafter there is no further disturbance and the form of the curve is
EF. This is the general case. The amount of the total disturbance
up to any instant is the difference in ordinates between the original
undisturbed curve ABG and the disturbed curve BEF^ such as H/
or GE.

For convenience, disturbances may be treated as of three sorts:

(a) Those disturbances which produce a finite effect in a single
instant; as, for instance, if the curve took the shape ABGEF,

Q>) Those disturbances which vary during the total time of dis-
turbance, so that if the total time be divided up into an infinite
number of equal infinitesimal time intervals {dtn^y aJ^ infinitesimal
part {dbu) of the total disturbance ($u) will occur during each d^A,
and the total disturbance may be represented as:

5u = J d^, or as

^'L ^^"

where ^ is the time the disturbance starts and T is the time it ends.
This is the most general case.

(e) Those disturbances ol which a proportional part ooours durii^
any part of the total time. These may be regarded as a special
case ot h,

ddu =^1 (ft A

where Cj is some constant.*

It is essential that one element of the trajectory be considered as
r^naining unvaried, so as to furnish a basis for measuring the varia-
tions of the other elements. Accordingly time will be selected for
this purpose. Both the^ time (t) of which the other elements (x, y^
*'* y'f ^" } y" i 6*^0 0^6 functions, and the time (^a) at which a dis-
turbance occurs, will be considered as unaffected by the disturbance*

Accordingly we can say that:

U =0

5^A =0

»■ ^i^^— ^■^^— ■ ^^m^^mt I ^— ^^^— ^^ M^^^— ■ ■■ ^ ■■■»■! ■■■■■ — ■■ ■■■■ ■■■■I ■■■ ■■■■ |i^.

I As kefe gLv«a, c is • spedftl case of k But It is possible to legud both h and a as special caass of e.
Thus, if tbe eoastant ci becomM the variable J^, we have case (; whereasi, if the timelntervBl T-^U
becomes inflnitesiiiial and a« rematns finite, we have case a.

20 GOtlBSB IN EXTBBIOB BAULISTIGS.

and, differentiating:

Let us first consider disturbances of the first sort. A particle is
moving through space according to some definite law of motion,
expressed by differential equations. At a given instant of time
(<A=^o) ft^ instantaneous disturbance takes place, which changes
the value of x, y, «', and y', or some of them. Thereafter, the par-
ticle proceeds according to its original law of motion, of course not
on a continuation of its original regular curve, but on another such
regular curve, also satisfying the original differential equations.

^^^..^ ^ The amount of the instantaneous changes
^^^'^'^""'^'^ in x, y, a:', and y' will be designated, re-
spectively, by 6ir, 6y, 6x', and hy\ Any one
of these expressions can represent either
^'^•^ a positive or a negative change. The

changes which will be considered in any practical appUcation of the
principles of this chapter, and the effects resulting therefrom, are so
minute in comparison with the elements affected by these changes,
that the following restrictive definition can be given: Sx, 5y, etc.,
are small finite increments of x, y, etc., and are so minute that
second and higher order terms (such as hhx or dxdy, for instance)
are of no consequence, as compared with x, y^ etc., and hence any
terms containing them may be disregarded and dropped from a sum
or series in which first order terms occur.

The operator 5 is the operator employed in that branch of mathe-
matics known as the calculus of variations, but not always with the
above restriction. No further understanding of the principles and
methods of this branch than here given is necessary to the purposes
of this book.

A certain resemblance between the operator h and the familiar
differential operator i will be noted. The distinction between the
two should also be noted. Consider a curve in an x,y coordinate sys-
tem. At any point on this curve, iy represents a continuous infini-
tesimal change in y along the curve, corresponding to an infinitesimal

change in x. ^ is the slope of the curve at the point in question.

hy and hx represent a very small fijiite break in the curve in question.
Ay and Ax may be considered as taking place in an infinitesimal period
of time (d^) ; whereas hy and hx may be considered as taking place
instantaneously, or as cumulating during a finite time interval.

III. EFFECT OF DIFFEBENTIAL VARIATIONS. 21

Let us now derive some of the basio theorems relative to the
operator 6.

8. Independent changes in u and u' may he made at any time tA-
Proof: Consider the curve AB in x,t coordinates, and W any

given point thereon. (See Fig. 2.)

The curve can be moved up or down, thus changing the x of W,
without changing its t or its slope. Or the curve can be rotated
about W, thus changing its slope, without changing its t or its x.

Thus 8x and d^ are independent. Q. E. D.

Similarly, by plotting y against t, and y against x, it can oe shown
that bxj bx', by, and 5j/' are all independent.

9. Terms containing more than one B may be dropped.

This was one of the fundamental hypotheses of the definition of
the operator 8 on page 20.

It will now be demonstrated that the four general formulas of
differentiation still hold true when the operator 6 replaces the opera-
tor d; in other words, that the small finite increments of this chapter
obey certain laws, already familiar in form for the case of differ-
entials, although, of course, these increments are quite different
from differentials.

These four theorems are as follows:

10. 5 (cu) = c 8u.

11. d (u+v)=du+8v.
13. d (uv) =v du-{-u dv.

— u 8v

13.

(U\_ V du — U

Only one of these (namely, 13) will here be proved, the proof of
the other three being reserved for problems.
Proof of theorem 13:

Take the expression -, and give u and v each an increment. Then:

\V/ V

-\-Bu u V 8u — u 8v

•{-8v V 'iP -\-v dv

EiXpand this fraction by dividing the numerator by the denom-
inator, as follows:

V du — u 8v V 8u — u 8v dv du u dv 8v

H -= — + • • •

V^-\-vdv V^ V^ V

22 COUBSE 137 EXTERIOR BAIJJSTIGS.

From the right member all tenns, except the first, may be dropped,
by theorem 9. Therefore:

(^^yvju^v ^^^

Next let us derive the expression for ''total increment," analagous
to the expression for ''total differential" (formula 2 of Chap. I).
14. Ifuis a function X and Y, then:

Proof: Let u=/ (X, Y). Give X and Y the increments ZX and
hY, respectively. Then:

hu^f{x+bx, r+5r)-/(x, Y).

Subtract and add the quantity/ {X, Y +hY), Then:

«u=/ {X H-«x, r +6 D -/ (X, r +« D +/ (X, r +5 y) -/ (x. y).

Applying the law of the mean (see Osgood, p. 230) to each of these
two differences gives:

Now, if these two partial derivatives are continuous, each would
approach the corresponding partial derivative of/ (X, Y) if bX and
iY both were to approach zero, and hence will differ but slightly
when bx and by are very minute.

Consequently we may express

a/-(x+g>ax, Y+sY) bf{x,Y) . dfix, y)

dX *^ dX "•"* bX

and

d/(X, Y+e,8Y) ^^ df(X, Y) . df(X, Y)

dY *^ dX +^ dY '

Substituting these values in the expression for du, and substituting
fj^forfiX, F), gives:

5u=»^-^ dX -{-jr^ 8Y +6X ^5rv'+^^ ^Kv

from which the last two terms can be dropped by theorem 9.
An extension of this derivation gives:

bu^^ 5X+^ SY+^ 5Z-f . . • Q. E. D.
16. ^ (*«)=*(«')

in. EFFECT OF DIFFEKEl^TIAIi VARIATIONS.

Proof: Consider a particle raoving along a curve from A to P.
l¥hen it reaches Py at time t^ let x and y instantaneously receive an
increment (5x and Sy, respectively) which will place the particle at
p, and thereafter let the particle move undisturbed along the curve
pD (the curves AP and pD being defined by the same differential equa-
tions of motion). Let Q be a point on the curve AP, such that, if
no disturbance had taken place at P, the particle would have reached
-Q after a small finite time
interval At from the time it
left P. Let q be the point on
the curve pD reached by the
particle, A^ seconds after
leaving p. Then a pair of
changes 8x + A8x and Sy +
A8y, occurring at time ^ +
Atf would produce the same
-effect as the pair of changes
^dx and 8y occurring at time i^. ^

Let 8x and 8y, although arbitrary at time ^, be thereafter con-
sidered as restricted by tne condition that at any instant thereafter
their value must be such as to produce at that instant the same
:situation as would have existed at that instant, had the changes been
made, with their initial values, at time ^. From this point of view,
the average rate of change of dx and 8y, during the interval At, is

respectively -rr and -^« Ai^ is here regarded as a change in the
lime of disturbance (^a)*

Adx =-BM-PG^GM-PR -^pr-PB;

Ady ^ij— Mr ^rq-- ML ^rq—RQ,

Let At approach zero. The rates of change at time t^ thus become
•dZx J d8y

FIG. 5

•dt^.

dti

d8x ,. A8x ,. pr—PR
(jUa At At

^==lim^=lim ^-^^
dt^ ^^ At ^^ At

Now, the unaffected aj-component of velocity (x') at time ti is the

lunit of

. . 'y as A^ approaches zero, A^ being here regarded simply as
.a change in the time (t) of which the elements of the trajectory are

24 GOUBSE IN EXTERIOR BALLISTICS.

functions. The value of %' at time ^ as affected by the variations, is
^« Therefore:

Similarly:

16. The cperaioT 8 does not always signify an independent iTicre-
fnent.

Examples: In the expression for the total increment given imder
theorem 14, if any three of the four increments are conceived of as
independent of each other, then the fourth must of necessity be
dependent upon the other three.

In equation 38 in Chapter IX, dX may be taken as a constant, and
8x as dependent on Sy, 5a;', and 8y'.

"niroughout this book, dt and dt^ wUl be taken as zero, by theorem 7.

So we may say that changes represented with the operator 5 are
essentially independent, unless rendered otherwise by some express
condition of the problem confrontiag us.

Let us now consider the following problem. If a particle, moving
according to some definite law up to time <a, suffers a disturbance
and then moves on according to the original law, without further
disturbance, what is the relation between the value of its x at some
later time Tand the value which x would have had at time TH there
had been no disturbance ? Let X represent the undisturbed value of
X at time T, and let X + dX represent the disturbed value. The
problem is to express 8X in terms of the 5a;, 8y, 5a;', and 5t/' occurring
at time ^a*

If y, x', and y' are not changed at time <a, then 8X is of the form
L 5a;, plus terms containing more than one 5, which can accordingly
be dropped by theorem 9. L will have a value which will depend on
the trajectory in question and on the poiat on that trajectory at
which the change 5a; occurred. Thus for any given trajectory L is a
function of tA, but is not dependent on 8x.

Similarly, if x, a;', and y' are not changed at time ^a, the resulting
8X can be expressed as M8y, etc.

If all four of X, y, a;', and y' are changed at time ^a, the resultiag 8X
will equal L8x+ M8y+ N8x'+P 8y', plus terms containing combi-
nations of 5a;, 8y, etc., which may therefore be dropped by theorem 9.

Thus we have, as the fimdamental equation for the X-effect at
time T, due to a set of small arbitrary changes in x, y, x', and y' at
time tAi

17. 8X ^L8x+M8y+ N8x' + P8y'

ni. EFFECT OF DIFFERENTIAL. VARIATIONS. 25

Formal expressions for Lj M, N, and P may be obtained from theorem
14, but have not yet been shown to be of any practical value.

The fact the Sx, 8y, 5aj', and dy' can each be given any arbitrary
value at any time of change (^a) enables us, if we wish, to assign any
arbitrary value we please to 6X and any three of these, and then
satisfy the equation by a proper choice of value for the fourth.

18. 8X is not a function oft^.

From the viewpoint of theorem 15, the values of 5a;, 5y, etc.,
although initially arbitrary, are regarded as restricted so as to change
value at such a rate, during the interval
dtiij as not to alter the resulting value of
bX. Q. E. D.

Thus fax, we have been considering
merely instantaneous changes. Let us
now consider the second sort of changes
listed at the beginning of this chapter,
namely changes which vary throughout^
finite time, starting as zero at the begin- ^ rlu -^

ning of the interval. A velocity change of this latter sort is readily
seen to be made up of an infinite number of infinitesimal changes
of velocity throughout the interval. Similarly a coordinate change
of this sort is seen to be made up of an infinite number of infinitesi-
mal changes in position throughout the interval.

Consider the finite time interval, from time t^ to time T, as divided
up into an infinite number of infinitesimal time intervals, dt^- An
infinitesimal part of the total Sx, dy, bx\ and by' occurs during each
of these infinitesimal intervals, and thus causes an infinitesimal part
of the total bX which will occur at time T. Thus the result, at time
r, of the disturbance during any interval d^., is

19. dbX^Ldbx+Mdby+Ndbx'+Pdby\

It should be noted that dbx, etc., are here used in quite a diflferent
sense from that of theorems 15 and 16. Here dbx means the change
which bx undergoes along the actual disturbed path of the particle,
during the interval dtA. But here also, d and b may be shown to
be commutative, as in theorem 15.

Consider x plotted against t Let ABG, Fig. 4, represent the t,x
curve of the undisturbed trajectory. Let a disturbance start at B
changing the shape of this curve to BED. Let EF ( « GH) represent
one of the infinitesimal time intervals, dt^.

26 COUBSE IN EXTBBIOB BAUilSTIGS.

The nonnal undisturbed rate of change of x is HI. The actual
disturbed rate of change of x is FJ, Thus:

Hl^dx

FJ--^dx + 8dx

Let EK be parallel to 01.

The total 5a; at the beginning of the dtA interval is GE. The total
8x at the end of the interval is IJ. Thus:

dSx^IJ^GE^TJ-IK^KJ
But:

ddx^ FJ-HI^ FJ- FK^ KJ

Therefore:

dhx

=<tD ^- ^- ^'

The right-hand member of formula 19 can now be transformed by
transposing d and 5, and by both dividing and multiplying each term
by dt^, and by applying this principle of commutativeness.

20. dbX^Lbx^dt^^ Mdy'dtA + NSx''dt^+Pdy''dtA
Integrating this from t^ to T, we get

21. 8X^1 L8x'dtA+ f Mdy'dtA-^- f Nhx^dU-^ \ PSy'^dt^

This is the formula for the effect, at time T, of changes which vary
throughout the finite time interval from t^ to T,

For solution of any of the integrals, the 6a?', 5y', 8x'' or 8y" therein
contained must be replaceable by something which is constant either
in value or in algebraic form throughout the interval.

Let us now consider the third sort of changes listed at the begin-
ning of this chapter, namely, changes which are proportional to
time. Consider a 8x of this sort. Then, from formula 19:

d8X=Ld8x.

Substituting dit^ for d5aj, and integrating within the limits of the
disturbance, we get :

8X^ I Ldt^

If L is a constant, this becomes

8X^L(T-t,).

in. EFFECT OF DIFFBRENTIAIi VARIATIONS. 27

PROBLEMS.

(19) Prove theorem 10.

(20) Pro ve theorem 1 1 .

(21) Prove theorem 12.

(22) Prove that 5 sin u==cos u 8u and that 8 cos u— —sin u 8u.
Suggestion: Turn to some book on the calculus, and follow the
Analogy of the similar differential formulas.

Questions on Chapter III.

1. What is the object of this chapter?

2. What is the meaning of the operator 6 in ballistics?

3. Do formulas involving the operator d hold true with respect to the operator 27

4. When would ^ represent the slope of a ciu*ve?

5. In this chapter what is meant by the word "trajectory"?

6. Distinguish between the two meanings of d6u and ddv\

7. What variations of time are considered in this chapter?
S. laSusji infinitesimal?

9. Distinguish between du and du,

10. Interpret Mof formula 17 by means of formula 14,

CHAPTER IV.

FINITE DIFFERENCES.

Integration by finite differences is based upon the principle that
inasmuch as a derivative is a rate of change, the value of an integral
of any smooth function can be computed step by step, if successive
values of its derivative are known at sufficiently close intervals.

Suppose that tt is a function of t, such that u can not be integrated
with respect to t by means of any of the expressions tabulated in any
table of integrals. In other words, formal integration is impossible.

Let us now imagine a plotted curve with u for its ordinates and t

for its abcissas. Then

dt is the area between the curve, the

/ axis, and the ordinates at ^=tn and ^— ^. Such as area exists for
every continuous curve, and hence even- continuous curve has an

integral, even though that integral
is not expressible by means of the
ordinarv elementarv functions.

In such a case, various approxi-
mations are possible, some crude
and some so refined that they can
t -4XW produce results to any desired de-
gree of precision.

All of the methods here described

lUt

FIG. 5

require that u be first tabidated for successive finite values of t
Let us consider the portion of the curve h ing between t^-i and ^k+j,

assuming the curve to be concave downward.
The entire required area can be divided up into sections such as

this. If we can evaluate each section, the sum of these evaluations

will be the value of the whole.

A first approximation would be the area under the chord.

Thus:

A =

'^k-l+'^k+l

V^+1 ~" ^k— l)

If we space t with unit intervals, then:

This approximation is too small.

A second approximation woidd be the area under the tangent at the
point (Uk, <k). Thus:

IV. FINITE DIFPEBENCBS. 29

If we space t with unit intervals, then:

23. A2==2u^

This approximation is too large, but the error is about one-half the
^rror oi A^. A, the true area, lies between A^ and J.,. Thus:

A^>A>A,

If the curve were concave upward, the above inequality would be
reversed. In either case it is evident by inspection that:

.^2A2 + Aj

^~~

Therefore a third, and very close, approximation is:

24. A^ « — ^ ^- - g (Uk-i + 4Uk + '«^k+i) .

This is known as Simpson's rule.^

PROBLEMS.

Evaluate the following to four decimal places, by formal integra-
tion, and by formulas 22, 23, and 24, and determine the percentage
of error in each case:

rio
(23) I x^dx

(24)

r dx

Jz X

In using Simpson's rule, when the tabular interval is other than
unitj^, the formula is:

25. ^s^'a (^'^)L-i+^k+'^k+i)

1 Formula 24 can be proved as follows: Any ordinary function of one variable can be expressed as a
power«eries of that variable. Thus the general curve which we have been considering so far in this
chapter can be expressed:

This equation represents a straight line, another straight line, a quadratic, a cubic, a quartic, etc.,
according as we drop all but the first, all but the first two, etc., terms of the right member of the
equation. For all approximations up to and including a cubic, formula 24 is precise. Proof: By this
formula.

Sh j^

tt(tt-^iio+4«h+tta,)
*

By formal integration.

rH^^^^f^ri

>2A

These are identical. With h sufiEldently small, similar pairs of expressions, involving quartics or
higher, are nearly mutually identical, being indeed coincident in as far as the first four terms are con-
cerned, the discrepancy in the higher terms being very minute. Q. E. D.
For another proof, see Osgood, " Differential aiii Integral Calculus," 1917, pages 406-408.

G0TJB8B IK EXTEKIOB BAIiLISTICS.

where h is the tabular mterval. The area of the next section will b&

A ('i^k-fi + ^k+a+'t^+s) Ai^d 80 on, so that the area for a series of sec-

h
tions will be o (^-1 + 4-0^+2^^+1 + 411,^+3 + 2^^+8 - • • 4u._i+uJ.

PROBLEMS.

(26) Divide I 2<2x up into ten sections, where 2 = x — x,. Tabu-
late z against x, and integrate by Simpson's rule. Integrate formally^
and compare the results.

(26) u is a function of t:

t^-10

(-0

Tabulate the values of u corresponding to ^»0, 1, 2 ... 9, 10.
rio
Evaluate I u dthy Simpson's rule and by formal integration, and

compare the results.

(27) Evaluate f TlOOO log io(lO+|^ j-1000 Idt, either by formal

integration, or by tabulating from a denary log table and then using
Simpson's rule.

Simpson's rule is a method of numerical integration, as distin-
guished trowi formal integration. It is one of the simplest of a
system of rules that may be obtained, involving the values of the
function and its diflferences. Let us now derive a method of numerical
integration which employs finite differences of various orders.

We will suppose a tabular fimction of t and will tabulate its first
second, third, and fourth receding differences as follows, when each
difference as tabulated is obtained by subtracting the element on
the line above from the element on the same line of the preceding
column, i. e., where %^fo—f^i] \ = cLQ — a^^] etc.*

fit)

First dif-
ference.

Second dif-
ference.

Third diJ-
terence.

Fourth dif-
ference.

-4

-3

«-«

-2

«-2

6-,

-1

a_,

6-1

«-i

(<.

s This rule should be followed regardless of the order in which the function is tabulated, i. e., regardless
of whether t increases or decreases down (or from left to right across) the page. Thus, if the tabular func-
tion of t algebraically increases as one goes down (or from left to right across) the page, the first difference
will be potUive; if decreasing, it will be negative. Similarly, if the first difference algebraically increases,
the second difference will be positive: if decreasing, negative, etc.

IV. FINITE dutbbbnges. 31

These differences are called ''receding/' for the reason that the
differences tabulated on any line would have receded if they had
been tabulated opposite the space between the two elements of
which they are tho difference; sometimes, for clearness, they are so
tabulated; but it is generally more convenient to tabulate them as
above.

(28) Successively evaluate /_i, /-a, /_^, and f^ in terms of /o, a©,
6o, %, and do-

(29) From this deduce a formula for/_n.

(30) Substitute t for — n. The result is the usual interpolation
formula in terms of receding differences, whereby / (f) can be calcu-
lated with values for any fractional value of t. The resulting/ {t) will
lie on a smooth curve within the interpolation interval and, if the
true/ {t) be smooth, will closely approximate it.

(31) What is the interpolation formula for t = i; for <= — i?

(32) In the tabulation of problem 27, what is the logarithm of
10.65 ? Use either interpolation formula from the preceding problem.

(33) Integrate the formula of problem 30, between the limits — 1
and 0.

(34) Integrate it between and 1.

(35) Integrate it between — 1 and 1.

The solution of the last three problems gives us, respectively, the
formula for integrating by finite differences and two formulas for
integrating ahead. These are the fundamental formulas of the
method of computing trajectories by numerical integration, which
method will be discussed in Chapter VIII.

(36) Form the differences of the tabulation in problem 26 and
integrate each interval '' ahead,*' using the formula of problem 34 in
the earlier stages, and that of problem 35 in the later stages.

(37) Integrate this same tabulation '' across," by using the formula
of problem 33. Compare the two sets of results.'

(38) Sum up the results in problem 37 and compare with the two
answers already obtained for problem 26,

s In numnical integration the values of the first few integrals will be inexact, due to the absence of first,
second, third, etc., differences in the integrand. To supply this lack extrapolate back for the missing dif-
ferences from such first, second, third, etc., differences as are obtainable from the tabulation. A better
method would be to rewrite the first four or five of the tabulated functions in reverse order, and difference
this new series as usual. What was the first interval being now the last, the set of differences to be used
in integrating over this last interval by the formula of problem 33 are now obtainable. The figures now on
the last line will be found to be the figures that were at the top of the columns of the original tabulation,
except that the signs of the differences of odd order are now changed. By the method here suggested the
student can construct for himself formulas for interpolating and integrating at the beginning of a table
with these adva-ncing differences. When this is done, the rewriting of the function in reverse order may be
discontinued. One of the methods of procedure outlined in this note should be followed in this and in all
suooeedtng problems of numerical integration. See Supplement H.

COUBSE IN BXTBBIOR BALLISTICS.

(39) Substitute 1 for t in the interpolation formula of problem 30.
The result is an extrapolation formula.

(40) Using the values /.j =/q — a©, /« =«/o, and the value of /i

obtained by the preceding problem; evaluate I fdt,j fdt, and
/ dt by Simpson's rule, obtaining the values of /-j and /j by

the formulas of problem 3 1 . Compare the resulting formulas with
those obtained by problems 33, 34, and 35.

(41) Derive formula 6 used in problem 18 of Chapter II.

(42) The last tabular values of t, y, a, 6, and c for a time interval
of one second are

-23.0

-2501

-8.2

+25

Find the value of t corresponding to y=0, using the formula of
problem 41. Interpolate for y as a check, using the formula of
problem 30. Carry t to four decimal places.

(43) The following tabulated values of y'' and x" from an actual
trajectory computation are given: 8-inch rifle, railway mount,
model of 1917. Muzzle velocity (F), 594.36 m/s; ballistic coefficient
{0)j 4.0; angle of departure (<^), 10°. The meaning of these terms
need not be considered at this stage.

Values of x'* and y" at two-second intervals:

(

-36.8

-16.3

-29.4

-13.9

-23.6

-12.2

-18.9

-10.9

-15.1

- 9.9

-11.9

- 9.3

- 9.4

- 8.9

- 7.5

- 8.6

- 6.3

- 8.4

- 5.4

- 8.2

Obtain y', x', y, and x for each value of t by numerical integration,
checking the final results by Simpson's rule. Initially x = 0, x'= F
cos 4>] yr=^Oj y'== Fsin fp.

IV. FINITE DIFFERENCES. 33

(44) From this data, calculate values for t, x, x\ and y' corre-
sponding to j/ = 0. As a check, get the value of y corresponding to
this value of t.

The formulas derived in problems 28 to 35 inclusive, and in prob-
lem 38, are to be used ordy with receding differences, and should be
plainly so marked in the student's notebook.^

Questions on Chapter IV.

1 . Define ' * integration by finite differences. ' '

2. State "Simpson 'a rule.''

3. How are first, second, etc., differences formed in the case of a tabular function?

4. What is the distinction between "numerical integration*' and "formal inte-
gration"?

5. From theoretical considerations rather than from the numerical results which
happen to have been obtained in any of the numerical problems, what is your opinion
of the relative precision of the six methods of numerical integration given in this
chapter? State detailed reasoning.

6. State the three formulas for integration by receding differences.

7. When should each be used?

8. What is meant by receding differences?

9. Give two methods of integrating ait the beginning of a table.

< For methods of deriving advancing difference formulas, see Supplement H.
24647—21 3

CILNJTER V.
ELEMENTS OF THE TRAJECTORY.

From the point of view of the present state of ballistics^ the follow-
ing are the more important elements and featmres of a trajectory:

Trajectory. — This term, as here used in connection with computar
tions; embraces only the standard trajectory of the projectile, moving
in accordance with assumptions which may be laid down as follows:

(1) The earth is motionless. (The average effect of the earth's
rotation on gravity is included in the assumed value of g.)

(2) The gun and taiget are in the same altitude above sea level.

(3) The preassigned standard muzzle velocity for that type of gun
is actually obtained.

(4) There is no wind.

(5) The atmospheric density varies regularly with the altitude
according to the exponential law assumed and is standard at the
muzzle (15° C. = 59° F. 750 mm. of mercury, 78 per cent saturation;
1.2034 kg/m»).

(6) The action of gravity is uniform in intensity, is directed
towards the earth's center, and is independent of the geographical
location of the gun, gr = 9.80 m/s'.

(7) In the computation of standard trajectories, the velocity-
resistance factor, G (v), may be regarded as dependent only on the
velocity; the standard variation in elasticity of the air being ac-
counted for by the value of the ballistic coefficient adopted.

(8) The ballistic coefficient is a constant on the trajectory and is
as determined from experimental firing. This includes the assump-
tion that all precessional and nutational effects of the projectile
may be ignored in computation, and hence the trajectory as computed
lies in a vertical plane.

(9) The vertical jump of the gun is as assumed from observation.

(10) The atmospheric temperature (and hence elasticity) varies
regularly with the altitude according to the law assumed, and is
standard at the muzzle (15°C. = 59°F.),i

1 A question arises in connection with assumptions (7), (8), and (10) as to the use of the temperature
structure. The standard temperature structure adopted by the Ordnance Department is supposed to
approximate average conditions aloft. Firing conditions on different occasions can be accurately compared
only when reduced to the same standard temperature structure, and for convenience the above-mentioned
standard temx)erature structure is always used as the basis of comparison. Variations in range due to irreg>
uiarities in temperature aloft are computed by reference to this standard temperature structure. But the
O function used in the computation of trajectories for standard conditions is assumed to be dependent only
on the velocity. This amounts to assuming, for the trajectory computation, that the temperature is the
same for all altitudes. The O function was based on a number of actual firings which were not corrected
for variations in temperature. These firings were probably made under average surface conditions, that is,
for temi)erature approximately 15^C.=59*' F. The computed trajectories involve a ballistic coefiOdent,
C, which is so taken as to make the range check with firings corrected to standard temperature structure
aloft. The explicit introduction of variable* temperature in the original computations, while possible
for individual trajectory computations, is not regarded as important, and is not feasible in the case of the
ballistic tables.

V. ELEMENTS OF THE TRAJECTORY. 35

(11) The drift (including lateral jump) is as assumed from obseirva-
tion.

Coordinates (x and y). — ^The coordinates of any point on the
trajectory, measured in meters. The abcissas (x) are measured
along the surface of the earth and are positive in the direction of fire.
The ordinates (y) are measured vertically from the surface of the
earth and are positive upward. The origin is the muzzle of the gun.
In the development of formulas and in computation, this coordinate
system is treated as cartesian, the error being negligible. Note
the difference between this conception and the tangent-plane con-
ception of the Ingalls-Siacci ballistics. The present conception,
being based on a curved earth, obviates the necessity for correcting
for curvature of the earth.

Surface of earth, — ^A spherical surface passing through the muzzle
of the gun and concentric with the earth.

Muzzle velocity (F). — A fictitious initial tangential velocity of the
projectile at the beginning of its flight in meters per second. The
blast may continue to accelerate the projectile for some distance
beyond the muzzle; so that the ''muzzle velocity '* is not the actual
velocity at the muzzle, but is rather a fictitious velocity which, if it
occurred at the muzzle and if there were no blast, would cause the
projectile to travel on the same trajectory as that on which it
actually travels.

VelocUy {v), — Sometimes called ''remaining. velocity." The tan-
gential velocity of the projectile at any point of its flight, in meters
per second. The x and y components of the velocity are repre-
sented by x' and y' respectively.

Accel-eration (x" or y''), — The rate of increase of x' and y\ respec-
tively (in meters per second per second), at any point on the trajec-
tory.

Time {t), — ^The time in seconds elapsed in the flight of the pro-
jectile from the muzzle to any point on the trajectory, t is the inde-
pendent variable of the trajectory. (When considered as the time
when a disturbance takes place, the symbol is <a.)

Quadrant angle of departure (<l>), — The angle measured from the
horizontal to the tangent to the trajectory at the muzzle; sometimes
called the angle of projection.

Inclination (6), — ^The angle measured from the horizontal to the
tangent to the trajectory at any point on the trajectory.

Point of fall (under range table assumptions). — The point where the
projectile in its downward flight reaches the same altitude above sea
level as the muzzle.

{Geographical) range (X), — The distance in meters from the muzzle
to the point of fall, measured on the surface of the earth.

Quadrant angle of fall (w). — ^The. negative of the inclination at the
point of fall.

36 COUESE IN EXTERIOR BALLISTICS.

Retardation (R), — ^The retardation due to the resistance of air of
standard density and elasticity. 7? is a function of velocity and alti-
tude.

R = vE

E is called the resistance function; (? is a tabular function of v,
for convenience tabulated to argument -rxjr ; C is the ballistic coeffi-
cient; H is determined by the following exponential law:

* = .0001036

Quadrant elevation, — The angle between the horizontal and the
axis of the bore just before the gun is fired.

Vertical jump. — The algebraic difference obtained by subtracting
the quadrant elevation from the quadrant angle of departure.

Summit, — The highest point of the trajectory.

Maximum ordinate (^b). — The y coordinate of the summit.

Tim>e of flight (T). — The time (t) from the muzzle to the point of
fall.

Angle of site. — The angle whose tangent is the ratio of the difference
in altitude of gun and target divided by the range. The angle is
positive if the gun is higher than the target. Owing to the fact that,
in the present ballistics usage, zero altitude is considered as being
the same elevation above sea level as the gun, rather than as lying
in the plane horizontal at the gun, the angle measured directly with
a transit or level is only approximately equal to the angle of site,
although satisfactory for short ranges.

Point of splash. — ^The point where the projectile, in range firing,
enters the water.

Center of impact, — The average position of several points of splash.

Ballistic coefficient (C). — A purely empirical number, used in the
formula for the resistance function (E), C is a mean value (con-
stant over a given trajectory) of the reciprocal of the relative retarda-
tion. The relative retardation is the ratio of the retardation experi-
enced by the actual projectile to that which would have been experi-
enced by a certain fictitious '' standard projectile," moving at the
same velocity and altitude. C for any given projectile and muzzle
velocity varies only as a function of the angle of departure. C may
be represented as :

^"id'

where to is the weight of the projectile in pounds, d its diameter in
inches, and i the *' coefficient of form," so-called because it is largely

V. ELEMENTS OF THE TRAJECTORY. 37

dependent upon the form of the shell. This i is an empirical number,
its value being determined by giving the C in the equation above the
value necessary to make the range computed with that C equal to
the actually observed range, when the latter is reduced to standard
conditions.

Ascending branch. — ^That part of the trajectory in which the pro-
jectile rises.

Descending brancTi. — That part of the trajectory in which the
projectile descends.

Plane of projection, — ^The vertical plane including the line of pro-
jection, i. e., the tangent to the trajectory at the muzzle.

PROBLEM.

(45) Prove that— .

x^ = v cos 6

y^ = v sin 6

Questions on Chapter V.

1. State the eleven standard trajectory aasumptionB.

2. Which of these assumptions are always in error?

3. What corrections in practice have to be applied to compensate for variations
rom each of the ten assumptions?

4. Define Ej and each of its elements, and explain how each is obtained.

5. Draw a trajectory Ijdng in the plane of fire and label the following: Gun, surface
of earth, angle of departure, elevation, angle of jump, maximum ordinate, "range, point
of fall, angle of fall, ascending branch, descending branch, sunmiit. Such as have
symbols may be labeled by the appropriate symbol.

6. Draw another trajectory. Mark a dot on it to represent the projectile in flight
at the end of t seconds. At that point, draw an arrow to represent the velocity. Draw
arrows to represent its components, 7/ and y^. Label x, y, and 0.

7. Define each of the elements of the two preceding questions.

8. Why does the assumption of a curved earth (see trajectory assumption No. 1,
'^ Coordinates" and '' Surface of the earth") make it impossible to measure the angle
of site precisely with a transit or level?

9. Why is the assumption of a curved earth (see '* Coordinates") more convenient
than the assumption of a flat earth?

CHAPTER VI.
HISTORY OF EXTERIOR BALUSTICS.

The science of ballistics consists of three paxts, namely: Interior
ballistics, dealing with the behavior of a projectile in the gun; exterior
ballistics, dealing with its behavior during flight; and ballistics of
penetration, dealing with its behavior while entering the target.
This book deals only with exterior ballistics.

The fundamental difiPerential equations of 'motion of a projectile
in flight have been known since the time of Newton. These are based
on the two components of acceleration, which may be represented as'
follows:

Horizontal acceleration = —R cos 6
Vertical acceleration = — i? sin ^— ^7

in which R is the retardation due to the resistance of the atmosphere,
6 the inclination (to the horizontal) of a tangent to the trajectory,
and g the acceleration of gravity.

These equations look very simple, and would be if i2 were a constant
or were one of certain simple expressions, in terms of time, velocity,
X, y, or d. But R appears to follow no mechanical law which can be
algebraicaUy expressed. All that we know about R has been derived
empirically, and even the simplest available approximate expression
for R renders formal integration out of the question.

Two great obstacles to the development of exterior ballistics have
always been (1) lack of knowledge about R and (2) the difliculty of
solving the differential equations. On the basis of the methods pur-
sured in attempting to overcome these obstacles, the progress of
ballistics may be divided into three periods.

The first period, namely all the years prior to about 1865, may be
called 'Hhe algebraic period.'' During this period, attempts were
made to represent R by some simple algebraic expression, and thus
to solve the equations by artifices such as those found in the stand-
ard methods of calculus. At first it was generally assumed that the
motion of a projectile through the air satisfied the conditions laid down
by Newton, under which R would vary as v^. The experiments of
Robins and Hutton, and the two sets of firings at Metz, furnished the
first real data on R at velocities sufficient for practical ballistics. As
a result, R was variously supposed to vary as the square or as the
cube of Vj or as some combination of the two; and, based on these

VI. HISTORY OF EXTERIOR BALLISTICS. 39

suppositions, there were devised many extremely ingenious solutions
of the equations, all of which solutions were found in course of time to
be insufficiently approximate, and most of which were very laborious.
The second period, lasting from about 1865 to the beginning of the
present war, may be called ''the period of approximations." Practi-
cally all of the physics and mathematics of this period were based
upon Mayevski's formula:

in which R is the retardation, v the velocity, and C the ballistic
coefficient. It was assumed that velocities could be divided into a
few intervals (such as to 790 f/s., 790 to 990 f/s., etc.), each interval
with its own law of resistance, and each with its own constant value
for A and for n. The experiments of this period, notably those of
Bashforth and Mayevski, and the Meppen and G4vre firings, were
accordingly directed to finding how these intervals might be taken,
and what the proper constants would be for each interval. The
solutions of the differential equations of motion (notably those of
Bashforth, Mayevski, Zaboudski, Hojel, Siacci, Didion, Braccialini,
and Ingalls), attempted during this period, not only made use of the
Mayevski formula, but also, in developing the various equations,
used frequent approximations, most of them based upon the assump-
tion of a practically flat trajectory. Thus the equations wore finally
wrenched into a solvable form. Our familiar Ingalls tables were pro-
duced in this manner.

The third period, extending since the beginning of the World War,
may be called ''the period of numerical integration." Since the time
of Euler it had been known that, by the use of numerical integration,
the differential equations could be integrated in their original form
without resort to approximations. But the approximate methods
were always thought to be simpler and sufficiently precise, until
certain phases of long-range and high-angle fire introduced during
the World War compelled a resort to numerical integration. Thus
ballistics has reverted to the first principles.

The chief differences between the Ingalls-Siacci ballistics and
present ballistic methods may be summarized as follows:

The former used a datum plane tangent to the surface of the earth
at the gun. The latter uses the actual surface of the earth as datum.

The former assmned air of a uniform density, this density being
chosen at such an average for each arc as to give the correct range
difference for each arc, and hence the correct range for the point of
fall, but not giving the other terminal elements precisely, and pro-
ducing undependable results at other points on the trajectory, es-
pecially in high-angle or long-range fire. The latter assumes the air

40 GOUBSB IN EXTERIOR BAIJiilSTICS.

of a standard structure approximating closely to observed data: less
dense the further one gets above the surface of the earth, in accordance
with an explicit mathematical law.^

The former represented the trajectory by equations which had
been simplified by a step which assumed (for the purposes of this
step) that the trajectory is a continuous straight line. This assump-
tion produced grave errors in high-angle fire. The latter solves the
equations in their strict form as originally deduced from the laws of
physics.

The former approximated the observed atmospheric resistance
function by certain rather roughly continuous formulas. The latter
uses the resistance as a tabulated function, as derived from experi-
ment, and smooth throughout.*

The former found it very diflBcult to obtain explicit data without
assuming^ the rigidity of the trajectory, which of course is inadmis-
sible if the target is considerably above or below the level of the gun.
The latter is adapted to furnishing just as exact information concern-
ing any point on the trajectory as concerning the point of fall, and
hence need not assiune the theory of rigidity.

The former possessed no practical method of correcting the range
for variable wind aloft, or computing the weighting factors there-
for, and hence often gave results so incorrect as even in exceptional
cases to have the wrong sign. The latter can compute, with any
desired degree of precision, the effect of any atmospheric change
occurring at any point on the trajectory, on the basis of the usual
physical assumptions.

In addition, inodern ballistics frankly treats the ballistic coefficient
as being purely empirical; and also introduces two new corrections,
which become important in long-range fire, namely corrections for
rotation of the earth and for changes in the elasticity of the air.
The derivation of the present day formulas is, on the whole, as simple
as the derivation of the formulas of the Ingalls-Siacci system, and
their %ise in the construction of ballistic tables and range tables re-
quires only the most elementary mathematics.

In view of the foregoing comparison, it is obvious why the later
method is better adapted than the earlier to the high-angle fire, the
long-range fire, and the fire at a target considerably above or below
the gun (including antiaircraft fire), which predominate in modern
warfare.

1 Of. "Physical Bases" (Ordnance Textbook 972), p. 5.
« Cf. "Physical Bases'* (Ordnance Textbook 972), p. 4.

* This assumption is: To strike a target at a different level from that of the gun, a trajectory may, without
chariffinQ its form, be rotated vertically, about an axis through the gun. See page 68.

VI. HISTORY OF EXTERIOR BALLISTICS. 41

The numerical integration which characterizes modern ballistics
is often called the "short-arc method.'' Short arc methods have
existed in the past. That of Siacci (discarded by him) consisted in
carrying his approxiiriations only during a change of say 5° in 0,
then starting a new set of approximations based on the value of 6
at that point, and carrying this new set for the next change of 5° in
df etc. But the words "short arcs" and "successive approxima-
tions" should not lead the student to assume that Siacci' s method
of making a series of approximations over successive short arcs bore
any close resemblance to modern methods of numerical integration.

Numerical integration has long been used to compute the orbits
of heavenly bodies, but was applied to the computation of trajecto-
ries in this country for the first time in 1917. The necessity for these
more exact methods was realized in this country as the result of
reports from England and France in which analogous developments
had arisen.

The development of the present ballistic methods in this country
have been eovered in the introduction.

Let it not be thought, however, that present methods have rele-
gated the familiar Ingalls tables to the discard. The Ingalls-
Siacci methods were devised for guns which were rarely if ever fired
at an elevation of over 15°; therefore their failure to apply to higher
elevations is not to their discredit. For fire up to 8°, the French
still use the old methods, even as a basis for their new tables. Until
the new American ballistic tables are completed, there is no reasort
why the Ingalls tables should not be used for all low-angle short-
range computations.

And even in the computation of higher-angle, longer-range trajec-
tories by present methods, the Ingalls tables have an important
place, for by their means the observed -range of range firings can be
corrected to the range-under-standard-conditions, and the first ap-
proximate value for C can be obtained from this standard range,
the muzzle velocity and the angle of departure.

Then, too, the muzzle velocity is obtained from the observed in-
strumental velocity by means of Ingalls' tables, which have many
other practical uses in ballistic experimentation.

The object of the new methods is thus seen to be not to supplant
the Ingalls tables, as being something inaccurate and obsolete; but
rather to treat these tables as sufficient in the field for which they
were designed, but as needing to be supplemented in the larger field
of modern artillery fire, for which they were never intended.

42 COT7B8E m EXTERIOB BAIiLISTICS.

Questions on Chapter VI.

1. Give the dates and characteristics of the tfiree periods of ballistic development.

2. What have been the t'vro great obstacles to the development of exterior ballistics?

3. Define the three branches of ballistics.

4. Compare the present assumption relative to atmospheric density with that of
the preceding period.

5. Compare the treatment of the equations of motion.

6. Compare the assumptions relative to rigidity.

7. Compare the assumptions relative to atmospheric resistance.

8. Compare the treatment of the ballistic coefficient.

9. Compare the coordinate systems.

10. What new corrections have been introduced by present methods?

11. Into what three parts does the study of exterior ballistics naturally divide?

CIMPTER VII.
THE MOTION OF A PROJECTILE.

For the purposes of the computation of trajectories and differential
corrections, the projectile in flight is treated as a particle. The
motion may thus be confined to the plane of flre, and the various
effects of the oblique presentation of the projectile to the air (i. e.,
drift and the range effects of yaw) may be temporarily disregarded.
Drift is subsequently treated as an empirical deflection correction;
and the range effect of yaw will probably be treated as a differential
range correction when the gyroscopic action of the projectile has
been studied somewhat further than at present.

The atmospheric retardation of a projectile in flight in still air
depends upon three things, viz., the velocity of the projectile, its
physical characteristics, and the density of the air. Then the at-
mospheric acceleration (i. e., the negative of the retardation) can be
expressed as follows:

26. a = -jFT

where F is a tabular empirical fimction of v. This is the F of the
Siacci ballistics, and is not to be confused with the E (formerly
represented bj F) of present ballistic methods. II represents the
actual density in terms of standard density, and C is an empirical
constant (different for each projectile), employed to make the
acceleration correspond to the physical characteristics of the pro-
jectile.

By throwing all the effects of a change in velocity into F, then H
and G can be made independent of velocity. By regarding the
atmospheric density at the gun as constant, H becomes a function
of y (i. e., the altitude of the projectile above the level of the gun).
The exponential function (H= lO-^-'^^^y) has been found to be a
close enough approximation to the physical facts for all practical
purposes. is conceived of as an empirical function of the charac-
teristics of the projectile, and is constant over any given trajectory.

For standard density, H becomes unity. For the so-called stand-
ard projectile, C is unity.^ Thus F is the acceleration of a standard
projectile, traveling through standard air, at velocity v,

1 This sentence is to be taken as the definition of ''standard projectile." A standard projectile may be
of any form, weight, etc., provided only that its C is unity.

44 COUBSE IN EXTEBIOB BALLISTICS.

For convenience (as will later appear), F is replaced by vG, G
thus being the ratio of retardation to velocity of a standard projectile
traveling through standard air at velocity v. For further con-

venience, the argument of the G tables'* is ^nn'

Now it has been found by observation that the true G is practically
dependent upon velocity alone, regardless of (7, H, and d. The
very slight effects of changes in C, H, and d on the true G have
therefore been taken out of (z, have been treated as constant through-
out any given trajectory, and have been merged in the C of that
trajectory, leaving the G function a function of v alone.

Equation 26 becomes:

rt« 7v/i GH T,

27. a= ^= j^v = — hv

E being adopted merely as a convenient expression for -p- •

The curve obtained by plotting G against v, shows that this
function changes most rapidly around the velocity of sound
(v = 330 m/s) and apparently has an inflection at that point. This
suggests a relation between this function and the velocity of sound,
which leads to expressing G as vB, where B is a function of the ratio
between v and the velocity of sound. This relation is not accidental.
It has a physical basis which can be derived theoretically.

This gives G the dimensions of velocity. H has the dimensions
of density. Accordingly C has the dimensions of sectional density,
i. e., weight divided by length squared.' This suggests representing

C Qs-^y where w is the weight of the projectile and d its diameter.

But to make theory correspond to observation, it is necessary to
insert an empirical factor of ignorance (i) in the denominator, hence

Let us now resolve the a of equation 27 into horizontal and vertical
components, as follows :

2J. Jx" = ~ Ev cos 6= - Ex'

V'== -Evsme== -Ey'

But there is an additional impressed acceleration, namely, 'gravity
i — g). As this acts only vertically, equations 28 become:

( x"- -Ex'

29.

tan^==^

* Any other dimensions might be chosen for 0, H, and C, provided that they combine to give correct
dimensions to E. See supplement C.

VII. THE MOTION OF A PROJECTILE.

These are ^he equations of motion of a projectile at any point of
its flight, referred to cartesian axes horizontal and vertical, respec-
tively, at that point.

But the earth is not flat. Accordingly, cartesian axes, which are
horizontal and vertical at one point on the earth^s surface, will not be
so at any other. Furthermore, gravity decreases with altitude. The
question therefore arises, how to adapt equations 29 to the actual
state of the earth.

The most convenient ways would be as follows:

(a) '^The tangent method. '' Consider the x axis as horizontal at
the gun. The equations become (see supplement E) :

30.

X ==

y" =

^^'"1^+

tan

2m^

Where R is the radius of the earth, and g^ is 9.80 meters per second-
squared. All ranges, altitudes, and slopes are relative to the axes
rather than to the earth, and the range at least must be corrected so
as to relate to the earth, even in the case of short-range fire. But
for purposes of computation, equations 29 are a suflicient approxi-
mation to equations 30, except in the case of long-range fire.

(6) ^'The curved method.'' Measure x in a circle passing through
the gun and concentric with the earth. Measure y vertically from
this circle. The equations become (see supplement E):

31.

/.../

Ex'-

2x^
B,

■ «

.//

tan0

i^y 9^^ R ^ E^

2yx'^
R

x' V fi

• • •

)

All ranges, altitudes, and slopes relate to the earth. For purposes of
computation, equations 29 are a sufficiently close approximation
to equation 31, except in the case of extremely long-range fire.

(c) Trajectories could also be computed by a third method which
treats the x axis as tangent to the trajectory at the summit. This
is sometimes called 'Hhe secant method,'' for it makes the x line
through the gun secant to the earth.

46 COURSE IN EXTERIOU BALLISTICS.

All range tablos now computed are based on trajectories computed
by ''the uncorrected curved method," i. c., r is measured along the
surface of the earth and y is measured verticall}' from the surface
of the earth, but equations 29 are used as an approximation to the
more precise equations 31. The reason for adopting the curved con-
vention is expressed as follows in the Aberdeen instrtictions for
range firing :

''The l(»vel surface may be taken to be either the curved surface of
the earth or the tangent plane to the earth at the position of the gun.
For the purpose of the gunner, the former would be the more con-
venient in case the levels of gun and target were taken from a
contour map, and the latter in case the levels were determined by
sighting from the gun. As the difference would be insignificant
except at long range>s, where presumably the first method would be
emplo^'ed, it is the custom at Aberdeen to determine the range for
the curved surface.''

QUKSTIONS ON ("lIAPTER VII.

1. Why ifl the projectile treated as a particle?

2. Upon what three things does atmospheric retardation depend?

3. What is the ^* standard projectile"?

4. What are the dimensions of E, H^ G, and C ?

6. What are the relative advantages of the "tangent method" and the ''curved
method"?

6. How are the x and j/ of a modern trajectory computation considered to be
measured?

7. What are the equations of motion used in computing such a trajectory?

CHAPTER Vni.
COMPUTATION OF TRAJECTORIES.

[Rectangular method.]

In modem methods, a standard trajectory is computed from
-given values of <^, V, and C, by means of numerical integration (see
Chap. IV) and successive approximations (see Chap. II).

The following tables are used:

Logs and Yoo

Table of the Function.

Logio ff =- -0.0000451/.

Some computers considerit quicker to compute log^o H by subtracting

^ from ~, pointing off four places, and then subtracting from zero,

than it is to use the last mentioned table; accordingly the H table
may be omitted, if desired.

The following blank forms are used: ^

'' Trajectory sheet," Form 5042.
" Computing sheet, " Form 5041 .

X, x', x^^Vf y/ and j/^' computed as follows:

For each of these variables there are, on the trajectory sheet, four
blank colxmms, the left-hand column being for the variable itself and
the other Uiree columns being for the first, second, and third diflFer-
ences, respectively.

The initial data are: The muzzle velocity (F) in meters per second,

the ballistic coefl&cient (C), and the angle of projection (<^). On the

first line of the trajectory sheet enter the initial values of x, x\ tj y,

and y\ as follows:

x =0

x^ =V cos <t)

. < =0

y =0

y' = F sin <p

Then turn to the small computing sheet and compute the first
column, for ^ = 0. On this sheet, log a:'^ and log y^^ should be crossed
out and log v' (which is a misprint) changed to log y\ Colog C and
g are constants throughout the computation. Colog C is obtained

1 Alternative methods of much merit are given in Supplements A and G. Of the three methods, that
of Supplement G is the one at present favored by the computers of the Technical Staff.

48 (BOURSE IN PiXTERIOR BALLISTICS.

from the initial data; g is 9.80. On both sheets F should be changed
to Ej to conform to the approved notation. The logarithms are all
to base, 10.

The computing sheet is used to obtain x" and y" from the funda-
mental equations of motion, derived in the preceding chapter:

32.

The process is as follows: Taking the value of z' from the trajectory
sheet, look up at the same time log x' and --^ in the table of logs

and squares. Similarly look up log y' and -^.-tt-

Add ynn ^^^ 1 0?) to g6t \-7^' With the latter, enter the table,

and take out log O,

With y, as tabulated on the trajectory sheet, enter the H table
and take out log H; or compute log H from y without using the table.

Add log Oj log Hf and colog C to get log E,

Add log x' and log E to get log Ex\ Add log E and log y' to get
log Ey\

Look up Ex^ and Ey^ from their logs, in the table of logs and
squares. Be careful to give Ex' the same sign as x' and Ey^ the same
sign as J/'.

Add Ey' and g algebraically, to get Ey' ■\-g.

Change the sign of Ex' and of Ey' ^-g, and enter them on the tra-
jectory sheet on the same line as the data on which they were based.
The reason for this change of sign is that x" equals minus Ex' and
not plus Ex' , Similarly for y".

We are now ready to begin a new line on the trajectory sheet. Our
second t should usually be \ second. Enter this value in the '*Time''
column. Two formulas for integration ahead are available for
breaking into a new line. This first formula is :

33. p/d< = i(/t+^at + ^&t+|ct+^5t)

where t is the time of the last complete Une and i is the time interval.
For example, if /t represents the tabulated value of x" at time tj then
the integral lepresents the increment of x' during the interval from
t to t-\-i.

In proceeding from f = to < = i, we have no values of a, &, c, etc.,
corresponding to x", and so have only the rough approximation:

•i „ ,. 1

x" dt^l X,".

VIII. COMPUTATION OF TRAJECTORIES. 49

As this integral represents the increment of x' from time to time
i, its value should be entered in the a column corresponding to x'
for time J. Add algebraically to the value of x' for time 0, and enter
the sum as the tentative value of x' for time J.

Similarly, integrate ahead for a tentative value of if for J.

We have now broken into our new line, and are in a position to
get a tentative value for y for time J, by means of the standard
integration formula

Now return to the small sheet, and compute a new column, as
before, using the tentative values of x', y'j and y for <«i. Enter
the resulting values of x" and y" on the trajectory sheet on the line
for t = \j and int^ate (this time by formula 34) for x', y', and y.
Substitute these improved values for the tentative values previously
set down, and repeat the computation on the small sheet.

Continue this process of successive approximations until you
obtain values of x', x", y, y\ and y", which check throughout. Then
integrate for x by formula 34. The smoothness of the differences of
X is a valuable check on the accuracy of x'.

Then proceed, in the same way, to get the values of x', x", y, y',
y^', and x, for < = i, ^ = J, and i=l. Each succeeding hne will, of
cour^, furnish more of a, 6, c, etc., for use in formulas 33 and 34.

After completing the line for <=*!, it may be well, though not
essential, to extrapolate back for values for a, &, and c for f =0, and
repeat the computation up to < = 1 again. This results in smoothing
out the curve and giving greater precision to the initial steps. By
using this smoothing-out process, it is often possible to start with a
one-second interval ^ and obtaui as precise results as a quarter-second
interval would give without the smoothing-out process.

After completing the line for < = li, skip a few lines, copy down
the values for x, x', x", y, y', and y" for times 0, \, 1, and H, and
make up new columns of a and h corresponding to each of these, but
now based on a AoZf-second interval instead of on a g^uar^er-second
interval as before.

Whenever third differences are available, the following formula
for integrating ahead wiD be found much simpler than formula 33 :

/»t+i

36. I fdt=i[2ft + \ (&t + ct+dt+ • • 0]

Jt-i

It is to be noted that the^rs^ difference (at) does not enter into
this formula, and that the increment obtained is to be added to the
value for the ^preceding Une. For example, in integrating x" at time

* See Supplement G for a method of starting at one-second interval.
24647—21 4

50 COUBSE IN EXTERIOB BALUSTICS.

20 to get x' for time 21, add the increment obtained by formula 35

to the value of x' for time 19. The increment, if obtained by formula

33, would be added to the value of x' for time 20. When first using

formula 35, it would be well to use formula 33 too, and conipare

results, as a check.
Go on from this point at half-second intervals, remembering that

% now equals \, Continue the process imtil y becomes negative.

This means that the projectile has pierced the initial plane and has

passed below the point of fall.

It is well to carry x" and y" to hundredths, and x', x, y', and y to
tenths of meters throughout the computation.

Often during the computation, and certainly at its close, the total
increment of x, x', y, and y', from time zero, should be checked by
integrating x', x", y\ and y" by Simpson's rule, preferably using
some type of calculating machine.

Usually, in computing trajectories, a one-second interval is adopted
shortly after the start, the change of interval being accomplished
in a manner similar to that prescribed for changing to a half-second
interval.

Intervals greater than one second should not be used by the stu-
dent, for the reason that data at one-second intervals are frequently
necessary for the subsequent computation of the differential cor-
rections.'

To get the terminal values of ^, x, x', and y' (i. e., the values* cor-
responding to y=0), it was formerly the practice to use formula 6,
and the interpolation formula of problem 30. The following some-
what analogous formulas^ are simpler:

• The present practice of the Technical Staff (whether the method of this Chapter or that of Supplement
Q is employed) is to doable the interval every time the third differences in the doubled interval indicate
that fourth differences may be neglected. Accordingly, only very short trajectories are computed through-
out by a one-second intervalj the usual intervals at the end being two or four seoondB. The differential
corrections are seldom required to any intermediate time. When they are so required, it is less work for
the skilled computer to find them by interpolation than by retaining a smaller interval in the trajectory.

4 Taylor's expansion of y ^ "0 1^*7 ^ written strictly:

-•'•-«'..A.-»'-^'{»''+f[»"'+f (»"''+ ••••)]}■
Similarly:

In the approximate formulas of the text, ail terms in y"\ and higher derivatives have been dropped. To
get a closer approximation, make use of:

This gives us:

Ay//- (gofy'\ )Af

and similarly for to!* and Aa/, the other formulas all remaining the same.

VIII. COMPUTATION OF TRAJEGTOBIES. 51

Note that the M of this method equals iAt of formula 6. The
other deltas are the same m both methods.

Take the values of x, x\ x", t, y, y', and y" from the line in which
y has the smallest absolute values, and keep a strict accoimt of alge-
braic signs. Solve for A^ by successive approximations/ taking as
a first approximation:

y t

When A^ has been found, the terminal values are obtained as
follows:

' Ax'^x"tM

37.

Ax = (x't + ^)A<

ly'T = /t + Ay'

Skip a few lines, and tabulate the values corresponding to t=^T.
Compute the angle of fall by the formula:

tan a)*« — ^

PROBLEMS.

(46) Compute the standard trajectory which would result from the
following initial conditions :

F = 792.47 meters per second.
C=4.0

& The work on a slide rule is z& follows:
(a) Set the ruimer to y t on scale D.
(6) Bring y't on scale C up to the runner.

(d) Opposite (a of /'() on scale C, read the approximate Ay" on scale D.

(e) Opposite ^"4+^^ on scale C, read the approximate Aj^ on scale D.

(/) Bring y'»+^ on scale C up to the runner.

(ff) Opposite 1 on scale I>, read a new approximate At on scale C, compare with one found above.

(A) Repeat the process from (d) to (g), untU no further shift of the slide is necessary. Algebraic signs
must always be taken into consideration. Also, be careful to see that the various figures are cor-
rectly pointed o£F, so that, for example, a At that should be .043 is not used as .43.
Then At, Ay^\ Ay ', yt, (as a check), Az", Ax', and A x can be severally read on scale D without change of

slide. They will be respectively opposite 1, (a of i"t)/i, y"t+^» y't+^i (a of i"t)/<, i"t+^, and

*' -I- A*'

52 COURSE IN EXTERIOR BALLISTICS.

Start with quarter-second intervals. Do not use the smoo thing-out
process described on page 49. Change to half-seconds after ^ = 1^,
and to seconds after computing < = 3.

(47) Use the same initial conditions and half-second intervab.
After computing the line for f = 2, extrapolate back for a, &, and c
of x" and y", for t — 0, and recompute to t^2. Change to one-
second intervals at f = 3. Compare the results with problem 46.

Questions on Chaptbb VIII.

1. What two mathematical proceefiee are used in computing a standard trajectory?

2. What tables are used?

3. Briefly, just what is a ''standard trajectory"?

4. What initial data are used?

5. What equations of motion are used?

6. What are the formulas for integration ahead?

7. When is each used?

8. What is the standard integration formula?

9. How are the computations checked from time to time?

CHAPTER IX.
DERIVATION OF AUXIUARY VARIABLES.

A given ballistic coefficient, muzzle velocity, and angle of departure
determine a standard trajectory, all the elements of which can be
computed by the methods laid down in the preceding chapter. In
that chapter we saw that any given trajectory has, for each value of
t, a definite value for x, y, a:', y\ x'\ and t/'^ The object of the
present chapter is to derive three auxiliary variables, /x, f, and p,
which shall each have a definite value for each point on each given
trajectory. In the next chapter we shall see how the formulas for
the various range corrections can be expressed in terms of these
auxiliary variables.

In Chapter III we saw how the effect on z at time T, due to a dis-
turbance at time <a, could be expressed as in equation 17:

hX^L 8x+M8y + Ndx'+P hy'

This is a general expression relating to the motion of any particle
moving under any definite law and subject to a disturbance at one
instant of time. Let us now specialize this expression, as follows:
Let the particle be a projectile moving in accordance with the laws
of motion evolved in Chapter VII. Ijet T represent the time of the
point of fall of the standard undisturbed trajectory. Then X is the
standard range. Let X, Tim, v, and p respectively be L, Jf, Nj and P,
specialized by these conditions, /i is the A of the exponential expres-
sion for // (i. e., J?=6""**y).

X is thus the change in range due to a unit change in x at time
^a; Am is the change in range due to a unit change in y at time t^y
etc. Thus these variable conversion factors are seen to have an
important physical significance. It is obvious that since x does not
enter into the diflferential equations of motion of a projectile, a
change in x at any point can have no effect other than to shift the
value of X at all subsequent points by the same amount. Therefore
X equals unity, and the expression for a range change may be written:

38. 5X = 5x + /^/x by-\-v 5x'-hp kj'

The effect of acceleration changes will not be considered in this
chapter.

Strictly speaking, the new range (i. e. , X + hX) will not be the range
to the point of fall of the disturbed trajectory (i. e., the point where

G0UB8E IK EXTEBIOB BALLISTICS.

y»0^ but rather will be the range to the point on the disturbed
trajectory where t^T, The time of the point of fall on the disturbed
trajectory will be r+ d T, where 3 T is the change in T due to the dis-
turbance. The X effect, at time T, of the disturbance at time Ia
will differ from the x effect at time T+6Thj some proportionally
very small part of the very small quantity 6X, and hence this very
small part of 6X may be disregarded, by theorem 9.

In equation 3S, dX is independent of tA, by theorem 18. By
theorem 15:

r.<'«>=<5)-«»'

dt
Therefore, differentiating equation 38 with respect to t^, we get:

dfi

39. 0««x' + A^5y-|-^/i hi' ■^ — bx*-\'vhx"^'^^hy'-\'phy

Now, by theorem 14, since x" and y" are each a function of x', y\
and y only:

40.

^x" &x" bx''

^ "dx'*'' ^W^ '^W^

Let us designate tt by /*', and similarly for i>' and p'.

Substituting the values from equations 40 in equation 39 and col-
lecting the terms, we get:

41.

=(h^'

^ dx''dy"

)«y+(

dx" , by"

^ + >''+>'->:zr+P

dx' ^'' dx

7-)«x'

(ft/*

dx" dy"

r)V-

Now since Sy, Sx', and 8j/' were chosen independently at time h
the algebraic law of vanishing 'coefficients applies, and each paren-
thesis above equals zero, whence:

42.

V =

p' =

— V

dx" dt/''

-p

dy ^ dy

^ dx" dv

— 1 — I'-^r-r — p

dx'

, dx"

— nfl — V ->r—f- — p

dx'
dj/"

by' "iyy'

IX. DBEIVATION OF AUXHJAKY VABIABLES.

Let us now perfonn the partiat difiFerentiations indicated in equa-
tions 42. Since

t" Ex\ and

y"~-Ey'-g,

then,

43.

( dx"
dx'

dx"

dx^

dy'

by^

dx'

= -E-x

,dE
dx'

— X

'= — X

,dE

by
,dE

dy'
,bE

-ydi'

,bE

bjT^
\by'

= -y

^-E-y

,bE

by'

Let us now perfonn the partial differentiations of E, indicated in
equations 43. Since

E'=-7j-, and

v'=x"+i/", and

then,

44.

ibEH dG_E d^ d£^
bx'~D'bx'~6'dv 'b%'

bE bH E, ,„

= ^ . ^ . ^ = Ez'( ^ log \
G' dv ' bx' \ vdv J

by~C'by ~H
bE H bG_E dG bv_
iby'^CWCfdv 'by'

-hE

^E dG bv ^ „y d log G \
G' dv ' by' ^ \ V dv /

Substituting the values from equations 43 and 44 in equations 42,
we get:

45.

{n'=-E{x'v + y'p)

56 COURSE IN EXTERIOR BALIJSTICS.

These equations hold for any possible combination of 6a;, ^, bx',
and by' occurriig at any point on the trajectory.

In these equations, the auxiliary variables are, by their original
definition, functions of the time of disturbance, ta. The elements
^} ^' y y\ Gf and V are functions of t in the original trajectory com-
putation; but it is evident that their values, for insertion in equa-
tions 45, must be the values which they have at the point of disturb-
ance, i. e., at the point t^t^. This condderation becomes important
in the solution of these equations in the next chapter.

Equations 45 can be solved for any given trajectory by making use
of the terminal values of M; ", and p, which are as follows:

46.

cot CO

These values are true because, at the point of fall a unit increase
in y produces an increase in X equal to cot co, but a unit change in
x' or y' produces no change in X.

Auxiliary variables may in a similar manner be derived with re-
spect to time of flight (T) and maximum ordinate (ys). The three
sets are then distinguished by the subscripts 1,2, and 3. ^

bX = \bx + hyi^by -h v^bx' + p^by'

b r=. \bx + htx^by + v^bx' + PjSy '

5.78 =• \bx -h Thix^by + v^bx' + p^by'

Hereafter in this book (except in supplement D) the omission of
subscripts signifies that the auxiliary variables retlate to range
changes.

PROBLEMS.

(48) What are the formulas for \\, ju'j, i^'a, and p',?

(49) What initial, terminal, and constant values are there for X„
\\y etc. ?

(50) What are the formulas for \\, m's, v\ and p\ ?

(51) What initial, terminal, and constant values are there for Xj,
X'g, etc. ? (Note that here the terminus is the summit.)

The computation of differential corrections has been simplified by
. the discovery of the following so-called ^'new first integral'':

47, x' + ^Mi^' + v^x'' + pyV'' =

This may be derived as follows. Consider equation 38:

5a; + Ihtx^by + v^bx' + pfiy' = bX

IX. DERIVATION OF AUXILIARY VARIABLES. 57

Restrict the four independent variables (i. e., 8Xj 5y, Sa?', and 5y')
by the condition that they shall represent a change from point to
point on the same trajectory, in the same injfinitesimal time interval,
dt. This converts the operator 8 to the operator dj and makes the
range change equal to zero. Thus:

dx + lif^idy -f v^dx^ + pidy^ =

Divide through by dt Then:

48. x' + hfjL^y' + j^ix" + py =

The similar ''first integral" for -j-. y equals unity; for -^ it equals

zero.

The above derivation gives the following physical meaning to the
''first integrals," namely, that an infinitesimal change in the four
elements (x, y, x', and t/') in time dt, which change amounts to
shifting from one point to another of the same trajectory, has no
effect on range or maximum ordinate, and makes a proportional
change in the time of flight.

Now substitute — Ex' for x" and — Ey' —giov y" in equations 47.
The "first integral" thus becomes:

x' + hp,y' - E (x'vi -f y'p,) -gp.^O
or:

49. " x' + hny-\-n/==gp^
whence:

Differentiating :

^ x' + ^juij/ ' -f m/
Pi* :;;;

, x'' + hpi,y'' + hn,Y + n^''
Pi == :;

Substituting these values in the last one of equations 45, namely, in:
we get:

x"+ Wy" + ^1^1 y' + m"x = - ghn, + E{x' + h„,y' + m',) -gn,'y'(~-f^)'

whence comes the following equation :

58 COUBSB IN EXTERIOR BAIX.ISTIGS.

which is the equation used in computing the auxiliary variables.
This equation produces the value of /i^' and fi^ for each value of <a<
The values of p^ and v^ can then be found from the formulas:

"i^-^/Pi-;^^

The former of these two is derived from equation 47. The latter
is obtained from the first of equations 45.
Thus we have, as the three equations of the system:

\,'\ = 2hEy'^,+^E-hy'-gy'(^^^y^ m'. + 2 Ex'

50.

Pi - - (x' + hy'ni + ii\)
'» ~ x'**' Ex'

The method of solving these equations is given in Chapter XIV.

PROBLBH.

(52) Derive the corresponding equations for juj", p,, and Vj] and

Ms"; Ps) and Fj.

Questions on Chaptbr IX.

j/Tf a..

1. Why does the "first integral" -jr equal unity; and the "first integral'' ^

equal zero?

2. What is the physical interpretation of Xi, h/ti^ /ii, and pj?

3. What is the physical interpretation of the fact that Xj equals unity?

4. In this book what does the omission of a subscript from the auxiliary variables
signify?

5. The auxiliary variables are functions of what, on any given trajectory?

6. Of what are x, y, x^, y', etc., fimctions, on any given trajectory?

7. Is it correct to state that the values of x^ and y^ used in the computation of auxil-
iary variables are functions of ^^? Explain.

8. Upon what subjects of mathematics, discussed earlier in this book, is this chapter
based?

9. Does X-\-5X equal the range of the disturbed trajectory? Explain.

10. Why are the terminal values of /t> »'» and p as given?

CHAPTER X.
RANGE CORRECTION FORMULAS.

IN GENERAL.

We have seen in the preceding chapter that at each point (^a) on
any standard trajectory there is a unique value for each of the
auxiliary variables: Xj, /xj, j'l, and p^.

Xj represents the change in range caused by a unit variation in x,
and is a constant. Xj = 1.

hfii represents the change in range caused by a unit variation in y,

v^ represents the change in range caused by a unit variation in x'.

Pi represents the change in range caused by a unit variation in y\

As only range changes are considered in this chapter, the subscripts
will now be omitted, to simplify typesetting.

The total range change caused by disturbances may be expressed
by consolidating all the range effects of equations 38 and 21 into one
range effect, hX,

I dhx^h] fid8y-\- I vddx' +

I pddy'+ I v8x''dtA-h I pdy^'dt
u Ju Ju

to dU

where the t^ of any of the four bracketed terms is the time of the
instantaneous disturbance considered in that term, and the ^o of any
of the six integral terms is the time of commencement of the dis-
turbance considered in that term, and the ^o of no two terms need
necessarily be the same.

Any specialized range change will be represented as AXj with some
subscript, so as to conform to range-table usage.

In the formulas of this chapter the auxiliary variables fi, v, p, and
their derivatives are functions of the time of disturbance, ^a. The
proper value of .t', y\ E, Gy etc. (which are normally functions of t),
to insert in these formulas are the values for < = <a.

The formulas given in this book are all (with the exception of
some in Chapter XI) formulas for the increment in range due to non-
standard conditions. Therefore, to correct the observed range to
standard range, or the map range to range-setter range, the quantity
obtained from the proper formula must be algebraically suhtracted.
This is in conformity with an agreement between the technical staff
of the Ordnance Department and the chiefs of Coast Artillery and

COURSE IK EXTERIOR BAULISTICS.

Field Artillery that range tables shall tabulate the range effects of^
rather than the range correct ions for ^ all nonstandard conditions other
than height of site.

EFFECT OF NONSTANDARD MUZZLE VELOCITY.

At the muzzle,

x' ^ V cos

j/'= T'sin

A 1 hijs increase in V increases x' by cos 0, and increases y' b3'
sin 0. Therefore, for b F= 1 m/« :

hx^ =« cos
6j/' = sin

Therefore, from equation 51:

52.

AJY'v= {v cos 0-fp sin 0)t.-o

PROBLKM.

(53) Derive the alternative form:

\ tx /t.-o

from the last equation of equations 49.

EFFECT OF CHANGE IN </>.

By problem 22, b cos <t> equals — sin ^ 5^, and h sin ^ equals cos ^ 5^.
It is customary to base the correction on a 1-mil change in 0.

50 = 1 mil = -^Yg radians.

A 1-mil increase in decreases x' bv V
7/ by T^ Tfrfq- Therefore, for A^^l mil.:

sin0
"1619^

5x'==~T

r sin0
1019

, ,. COS0

and increases

Therefore, from e<]uation 51 :

53.

A A'

<t>

1019

(p cos <f> — p sin <^)t.-o

X. RANGE CORRECTIOK FORMULAS. 61

PROBLEM.

(54) Derive the alternative form:

^^^ ■" 1019 (cos *^ Ex' ) t .^

CHANGES IN ACCELERATION, IN GENERAL.

From formula 51, the range effect of variations in acceleration is
seen to be:

54. IX ^ Cv hz"dt^+ f p iy^'dti

We are interested only in such acceleration changes as are caused
by variations in i?, inasmuch as variations in x' or y' (the other factor
of x" and y'', respectively) are cared for by the terms in Sx' and dy' in
formula 51. Thus for the purposes of the present derivation we may
assume that x' and y' do not vary. Accordingly:

55.

Substituting from equation 55 in equation 51 :

5X= I ~y-~c'^~G Ti) ^(^ " + 2/ p) ^^A

Substituting m' for — J?(xV + yV), from equations 45, we get:

EFFECT OF NONSTANDARD BALUSTIC COEFFICIENT.

For a 10 per cent increase in O alone, formula 56 becomes:

A.Yc= r (-0.1)M'rf/A

57. AXo = 0.1[Mto-MTl

EFFECT OF NONSTANDARD WEIGHT OF PROJECTILE.

An increase in the weight of the projectile over the standard weight
will increase the ballistic coefficient {O) proportionally, and will de-
crease the muzzle velocity (F) in accordance with the formula of

interior ballistics,

bV bp

V p

62 GOUBSE IN EXTERIOR BAIXISTIGS.

where p is the standard weight of projectile, and n is one of the vari-
able coefficients of interior ballistics (approximately —0.3), not to be
confused with the G&xfre n of the temperature formula (equations 59
to 64).

A 1 per cent increase in the weight of the projectile will change the
velocity by 0.01 nV, which in turn will change the range by that,
multiplied into the range effect of a 1 m/s increase in V (formula 52) .
A 1 per cent increase in the weight of the projectile will also increase
C by 1 per cent, which in turn will increase the range by one-tenth
of the range effect of a 10 per cent increase in C (formula 57) . Hence:

58. AXp = 0.1 AXo-hO.Ol Tn AXy

For values of n, see lesson sheet P of the Ordnance School of Ap-
plication: "Tables for Interior Ballistics," Table A; or use the value
-0.3.

It might seem that n should equal —0.5 instead of —0.3; for if
the muzzle energy produced by a given charge of powder were con-
stant, regardless of the weight of the projectile, then:

l(pi-dp)(v+8vy^lpV'

whence:

2dV_ 8p

V p

But a change in the weight of the projectile causes it to stay a
different length of time in the bore and alters the friction between
the gim and projectile, so that the muzzle energy is not constant.
Empirically, —0.3 has been found to give more nearly correct results
than —0.5.

In range-table firings, when the actual velocity is known, and what
is desired is the standard range, one range correction is made (by
formula 52) to correct from actual velocity to standard velocity, and
one to correct from actual weight to standard weight, using only the
first term of formula 58. The last term of formula 58 is not used,
for formula 52 covers the whole matter of velocity.

If the velocities are taken on special velocity rounds, rather than
on the range rounds, the velocity of the range rounds is obtained by
adding 6 F to the velocity of the velocity rounds, 5 F being found by
the formula :

5F 8p

V T)

X. RANGE OOERECTION FORMULAS. 63

where 5p is the difference obtained by subtracting the average pro-
jectile weight of the velocity rounds from that of the range rounds.
In service firings (or in any case where the range obtained by one
weight of projectile is known; and what is desired is the range to be
expected of another weight, all other things being unchanged) the
whole of formula 58 should be used.

EFFECT OF NONSTANDARD TEMPERATURE.

A 1 per cent change in absolute temperature alone will now be
considered, merely from the viewpoint of its effect on the elasticity
of the air.

Let 8 represent the velocity of soimd, s© the standard velocity of
sound, r the absolute temperature, and r© the standard absolute
temperature for the altitude in question. A normal temperature
structure has been adopted by the technical staff of the Ordnance
Department, based upon the normal temperature which occurs at
each altitude up to ten thousand meters, when the temperature at
the ground is standard. This temperature structure is incorporated in
a table, giving

with t/ as an argument. At the ground r© equals 518.4° in Fahrenheit
units, corresponding to 59** Fahrenheit.^

Kepresent G as the product of the velocity by a function of •

Call this fimction JB ( - V Then :

O...B(l)
het 8 = SQ-\-ds. Then:

^=^^CTTii)

Expanding by Taylor's theorem:

G=^v By-- I— v^ -1 +higher powers of —

\8q/ dV Sq Sq ^

59.' B6^0-Go---v'^^f

It now remains to evaluate -r- and ~

dv So

I See "Ph3rsloal Bases" (Ordnance Textbook 972), pp. 12, 13. Also note, p. 34, ante.

64 GOTJBSE IN BXTBRIOR BAIXISTICS.

60. G^vB''^

(N. B.— This ^ia not to be confused with the ^ of modem ballistic methods, vi^ch
n many of the earlier papers was written F.)

61. B^A^v""'^
Differentiating equation 61:*

Differentiating equation 60:

dB joiQ-G AvO /v dG ^\Q / v d log G ^\
dv v^dv v'^ \G dv J v^\ dv )

Therefore the GfirVre n may be defined by the equation:

The object of the above differentiations was to get n into the
form of a logarithmic derivative of G,

Now to evaluate - . In the vicinity of So; « varies as V^. There-
to

fore:

So 2to
Substituting from equations 62 and 63 in equation 59:

^G_jo^dJBh8__ hr

A 5r of 1 per cent would make:

-^ ===-(" -2)200-
Substituting in equation 56 :
64. AXt--0.005| (7i~2Vd«A.

t An and nare here taken, not as constants over certain intervals of o (of. Chap. VI), but rather as |

smooth slow-varying functions of v. But they vary so slowly, as compared with v, that they may bo |

4 roated as constants in diflereDtiatiiig.

X. BANOE OOBREGTION F0RMT7LAS. 65

A 3t of 1° Fahrenheit would make:

-g--(n-2)2^;
Substituting this value in equation 56 :

65. ^Xr - - r ^^^M'd^A.

Instead of using the tabular value of r^ in computing ^Xr, the
present practice is to use the ground value (518.4*'); and then treat
^Xt as though the value aloft had been used. This produces a slight
«rror, which is probably less than the errors in our knowledge of the
exact effects of temperature changes.

Formula 64 is simpler to compute and (as used) is more precise
than 65, and a percentage change is simpler to use with a ballistic
temperature; hence formula 64 will generally be preferred.

EFFECT OF NONSTANDARD DENSITY.

The weight of 1 cubic meter of air under standard conditions
<59® Fahrenheit, 750 mm of mercury pressure, and 78 per cent
saturation) is 1,203.4 gm. The values 59 ^^ and 750 mm were chosen
by the Technical Staff for the reason that these values conform as
nearly as feasible to the general practice of American, British, French,
and ItaUan ballisticians during the World War. Seventy-eight per
cent was chosen to conform with the present practice of the Coast
Artillery. Ingalls's Table III also uses 59®, but uses 752 mm instead
of 750 nmx, thus giving a density of 1 ,206 gm per cubic meter. Enter-
ing this table with 59® and 750 mm (i.e., 29.53"), one gets 1.002,
which for practical purposes is indistinguishable from unity.

The exponential law (H^e-^^) was derived on the assumption
that £* varies as the density. And, of course, density varies as weight
per any unit. Therefore H, Ej density and the weight of a cubic
meter at altitude y are proportional. Whence:

bH 5 (weight of a cubic meter)
•'• H " 1203.4 H

At the ground, H is unity. For an increase of 100 gm in the weight
sof a cubic meter of air at the ground:

^--^^=0 0831
H 1203.4 "-"^^l-

24647—21 5

66 ' OOXJBaB IN SXTBBIOR BALUSTIGS.

Substituting in equation 56 :

66. A-Yh - 0.0831 r /i'dtA=» 0.0831 O^t-mu).

Changes in atmospheric density aloft are converted into equivalent
changes in atmospheric density at the ground, by the formula :

^« Change aloft . , ^ j t_

67. % — equivalent ground change.

For an increase of 1 per cent in atmospheric density.

-g-0.01.
Substituting in equation 56:

XT
M'd<A = 0.01 (mt-mu)-

It should be noted that this is the same formula as that for AXc
(formula 57), except that it is based on 1 per cent instead of 10 per
cent and is opposite in sign.

Both the formula for 100 gm increase and the formula for 1 per cent
increase will be used in the computations of Chapter XVI.

EFFECT OF A REAR WIND.>

To determine the effect of a wind moving in the same direction that
the projectile is moving it is convenient to compare the motion of the
projectile relative to the air and its motion relative to the earth. In
other words, it is convenient to regard the motion of the projectile as
though the projectile were moving along a trajectory relative to the
air and as though this trajectory were itself moving as an entirety at
the velocity of the wind.

If a 1 m/s wind is blowing in the direction of the x axis from time
^ to time T, the x component of the velocity of the trajectory relative
lo the air at time to and thereafter will be 1 m/s less than the x com-
ponent of the velocity of the trajectory relative to the earth at that
instant, which amounts to considering the x' of the projectile as re-
ceiving at time t^ an increment (5x') of — 1 ; but the trajectory relative
to the air is itself moving as an entirety at a rate of 1 m/s from time t^
to time T. Thus the effects of a 1 m/s wind are two : (a) A minus 1 m/s
Bx' at time t^; (6) a plus horizontal movement of the entire system
at a rate of 1 m/s for T—to seconds. From formula 51, we take the

terms:

Jr»T
I ddx + vbx'
to

t Compare ''Physica] Bases" (Ordnance Textbook 972), pp. 11-12.

X. RANGE COBRECTION FORMULAS. 67

Now as, by (b) above, 6x is changing at the same rate as t^,dBx^dtA
By (o) above, fi^'tQ— — 1. Therefore:

1 m/s wind blowing throughout the trajectory:

69.

For a

* EFFECT OF A VERTICAL WIND.

Similar reasoning applies to the effects of a 1 m/s vertical wind«
(1) A &y^ of minus 1 m/s at time <»; (2) an upward movement of the
entire system at a rate of 1 m/s for T— to seconds. There are no
variations in x or x^

The effect of the decrease in y't^ is :

The effect of the upward movement of the system is:

XT
hfi d&y.

But, as 5y is changing at the same rate as t^, dBy^dty, hence:

£T
itdt^.

Thus the tofal range effect of a 1 m/s vertical wind, blowing from
time ^0 to time T, is:

71. AX^y^Tij ndtA — pto*

For a 1 m/s wind blowing throughout the trajectory:

/»T

72. AX^j =7i j jxdt^ — pQ.

CURVATURE OP THE EARTH.

In the Ingalls-Siacci system of ballistics, the practice was to com-
pute a trajectory by means of rectangular-coordinate equations of
motion based upon the assumption of a flat earth, and then apply the
trajectory to the ciu*ved earth by referring it to a rectangular grid,
whose X axis was tangent to the earth at the gun. Of course, in such a
system, the farther one got from the gun, the farther the surface of
the earth dropped away from the x axis; thus ciu*vature of the earth
increased the range by AXk^ 0.07848 R^ cot «, where R was the
radius of the earth.

68 OOX7B8B or EXTBBIOB BALUBTICS.

But the present convention is that the trajectory, computed exactly
as before, is now applied to the curved earth by referring it to the
orthogonal curvilinear grid discussed in Chapter VII.

Under the tangent system, curvature was taken out of the range-
firing range to get the tangent range, all range-table computations
were based on the tangent range, and then the curvature correction
had to be added in again.

Under the curved system, the curvature correction does not have
to be either taken out or put in. Inasmuch as the system is curved,
curvature of the earth may be disregarded.

SITE AND COMPLEMENTARY SITE.

In the Ingalls-Siacci system of ballistics, the practice was to
treat a difference in altitude between gun and target, by assuming
that a trajectory whose chord inclined from the gun to the target
would have the same shape as though its chord were horizontal.
This assumption was called the *' theory of rigidity of the trajectory."
It was defined by Col. Ingalls as assmning 'Hhat the relations existing
between the elements of the trajectory and the chord representing
the range are sensibly the same, whether the latter be horizontal or
inclined to the horizontal."

It is obvious that, all other things being equal, a projectile will
go farther if the ground slopes down from the gun, and less far if the
ground slopes up from the gun, than if the ground were level.
Assuming rigidity, and an angle of site (i. e., the inclination of a line
drawn from the gun to the target; positive if up) of € degrees, the
correction to target range would exactly equal the error caused by
a change of € mils in </>.

AX, = [AX^] «0-«

But the theory of rigidity is not sufficiently precise for some
phases of modem fire (see Gunnery for Heavy Artillery, pp. 66-69;
Gunnery for Field Service, pp. 23-25). The error in the theory of
rigidity was formerly corrected for by expressing the effect of this
error in terms of the change in elevation necessary to overcome it.
This correction was known as the ''complementary angle of site,"
the words ''complementary angle" obviously not being used in their
geometrical sense. The corrections for curvature, site, and comple-
mentary site were often combined into one correction.

The present practice is to derive a single range effect from the
coordinates of a computed trajectory. The following formulas are
used:

tan €= -

X. BANGE COBKECTIOK FORMULAS. 69

73. ^X ^X-X

The procedure is to plot c against x—X for each of the last few
computed points of the trajectory, draw a smooth curve, and from
it build up a site-correction table.

Whether the correction based on these formulas be called ''the
correction for site" or "the correction for site and complementary
site/' it covers dH of the effects of a difference in altitude of gun
and target.

PROBLEMS.

(55) Prove that—

56r == — v* T

dv «o

See the foregoing derivation of formula 59 for effect of nonstandard
temperature. Suggestion: by performing the indicated division,

(v \ • V

' . ) into two terms, one of which shall be — • By

Taylor's Theorem for a single variable, expand B ( . j into a

series of JS ( — ) and its derivatives.

(56) A shell weighing 15.96 pounds attains a range of 5,000 meters.
A 10 per cent increase in ballistic coefficient would increase that range
by 164 meters. A 1 m/s increase ifi muzzle velocity would increase
the range by 2.6 meters. What range would be expected of by a
projectile weighing 15.5 pounds, the powder charge and angle of
departure remaining the same ?

(57) A shell weighing 16.1 pounds is fired at the same elevation
and in the same gun as that of the preceding problem. The muzzle
velocity is measured and found to be 20 m/s above standard. What
range should be expected ?

QuBSTioNS ON Chapter X.

1. What is the difference between AX and dX?

2. Give a complete tist of the range effects discussed in this chapter, and state as to
each the unit change on which it is based.

3. How does the correction for weight of projectile in range-table firings differ from
the correction used in service firings, and why?

4. Between what limits should the formulas of this chapter be integrated?

5. What are the proper values of x^, i/, E, G, etc., to insert in the formulas of this
chapter?

6. How is curvature of the earth corrected for?

7. Is complementary site included in the correction for site?

8. Given that AX*^, »50, map range=» 10,000, what range would you set for a vertica 1
wind of 4-2 meters per second?

9. Would it be possible for A.Y, to equal — 25? Give reasons.

CHAPTER XI.

ANGLE OF DEPARTURE CORRECTION FORMULAS.

EFFECT OF JUMP.

Vertical jump equals angle of departure minus angle of elevation.
It is therefore positive if up. The gun is laid by the angle of eleva-
tion, but it is tiie angle of departure which controls the shape of the
trajectory.

Jump obviously affects the range through the formula for AX^.
The object here will be, however, not to evolve a formula for the
range effect of jump, but rather a formula for measuring the jump

TRAJECTORY--^
LINE OF VtFWCntRt^

JUMP

-(N^J) COSE'

SCREEN

FIG &

itself; inasmuch as it is ^, rather than X, which is always corrected
for jump.

A special blank form is used at Aberdeen Proving Ground for the
computation of jimap. Most of the symbols.used conflict with other
symbols of ballistics, but are nevertheless here given just as they
appear on the form.

8 is the horizontal distance, in feet, from the muzzle to the jump
screen.

E is the elevation at which the gun is laid for the jump firing.

D-^Ssec E.

J is the measured vertical error, in inches, at the jump screen, +
if up.

8 is the distance from the trunnion axes to the muzzle.

t is the time of flight from muzzle to screen, i. e., approximately -y'

N is the fall, in inches, due to gravity-g fl^'^^o ^^^-^^ ' 12 • P) —
193 fi.
70

XI. ANGLE OF 0&PARTITKB GORBECTIOK FORMULAS. 71

The pivot of jump has been empirically determined to be about
halfway from the trmmions to the muzzle, rather than at the trun-
nions, jump being due to a combination of rotation about the trunnions
and other motions, principally recoil.

j is the angle of jump.

. . (iV+cT) cos (E+J)
am 7 «= - — —

("-i)

the reasoB for the 12 being to reduce (D^^ ) *^ inches like {N+J).

But j is so small that cos (E-hj) is practically cos E, and / (in
minutes) is practically ^p- Sin 1' is ^^. Therefore

„. .^34 38 (N+J) cos E 573 (N+J)

' 12(D+t) -^(20+8) sec E'

EFFiilCT OF CANT.

Although the most marked effect of cant (i. e., trunnions out of
level) is the effect on deflection, yet it also has a slight effect on the
angle of departure.

See the "effect of cant" in Chapter XII. From the development
th^re, it is evident that cant has the following effect on angle of
departure:

A<^ (in mils) = - 1019 cos i (tan <^- tan £)

Converting into minutes,

75. A0=» -3438 cos i (tan 0-tan E^)

The i of these formulas is the angle of cant, and has no relation
to the form factor i, nor to the i of the jump formula.

When laying a gun with a quadrant or with a sight shank that
is capable of being leveled, the indicated elevation is the true eleva-
tion, and no range correction for cant need be made. Even in the
case of guns laid by range drum or by nonlevelable scale, the correc-
tion is apt to be inappreciable imless the tnmnions or base ring are
badly out of level.

ANGLE OF SITE.

The point of splash in range firing is of course at a lower level
than the gun. The exact difference may be ascertained by com-
paring the height of the trunnions above mean low water and the
height of the tide.

72 OOUBSB IK EXTBBIOB BAIUSTICS.

The difference will be so slight that the ''theory of rigidity ''^
(see p. 68, ante) may be assumed without necessitating any ''com-
plementary" correction.

This difference, divided by the range (being careful to convert to
the same units), gives the tangent of the angle of site. (See the
definition of "angle of site" in Chapter V.) As small angles are
approximately proportional to their tangents, and as the tangent
of 1' is -stVr, then the angle of site (in minutes) is equal to the above
ratio multiplied by 3438.

This angle must be added to the quadrant angle of departure to
give the ^ which would have produced the same X, if firing on the
level.

IN GENERAL.

It is often convenient to know what elevation correction is necessary
to offset any one of a number of nonstandard conditions affecting^
range.

Suppose, for instance, that the disturbing cause is a 1 m/s helping^
wind. This is found, by formula 70, to produce a certain AX.
Find, by formula 53, what A<^ will produce AX, numerically equal
but opposite in sign. This A</> is the desired correction.

Similarly can be found the elevation correction necessary to offset
any other disturbing nonstandard condition.

PROBLEM.

(58) A gun is laid at an elevation of 2^. From the muzzle to the
jump screen the horizontal distance is 100 feet. The length of the
gun from trunnion axis to muzzle is 20 feet. The shell pierces the
jump screen 30 inches above the point of bore sight. The muzzle
velocity is 2,000 foot-seconds. What is the jump ?

Questions on Chapter XI.

1. Why is the pivot of jump considered to be halfway between the trunnions and
the muzzle?

2. Define vertical jump.

3. Define cant.

4. Define angle of site.

5. How do you find the elevation correction necessary to offset the range effect of
some disturbing cause?

6. In deriving the jump formulasi why is g taken as 32.16, instead of 9.80, as in
the rest of the book?

CHAPTER XII.

DEFLECTION FORMULAS.

EFFECT OF CANT.

Cant occurs when one trunnion is higher than the other. The
tangent of i, the angle of cant, is obtained by .dividing ''right wheel
above left'' by the ''dis-
tance between levels.''

Suppose a gun is ele-
vated E degrees and bore-
sighted at a point on a
jump - screen 8 + s feet
from its trunnions, and
then is slowly elevated to

4> degrees. (See Pig. 7.) SIDE VIEW OF GUN /WD SCREEN (TRUNNIONS LEVEL)

If the tnmnions are level, FIG. 7

the line of bore-sight will

travel vertically up the screen a distance equal to

(S-hs) (tan </>-tan E).

Notice that in the case of cant the pivot is at the trunnions. But this

\^,-.RIGHT ABWE LEFT
-^^^--DIST/^NCE BETWEEN POINTS

^...

GUN

REJ\R VIEW OF GUM
(TRUNNIONS INCLINED) D

PIG 8 RE^R V'f W OF SCREEN

FIG 9

vertical line will be tilted i degrees to the left, if the trunnions are tilted

J. through i degrees by raising the
right trunnion above the left.
(See Figs. 8, 9, 10.) The line of
bore-sight will now cut the screen
(S -t- s) (tan <^ — tan E) Bin i feet
to the left of its original position,
and hence will have a deflection

-5*5 —

GROUND PL/IN
FIG 10

whose sine equals

(8 + 8)

(tan </>— tan E)

sln^, 1. e.,

(8 + 8)

sm i? = — (tan <^— tan E) sin i,

the minus sign being introduced because, deflection to the right being
taken as positive, and " right wheel above left " being taken as posi-
tive for cant, a positive cant produces a negative deflection.

74 COURSE IK EXTEBIOB BAUjISTICS.

Now, since small angles vary approximately as their sines, and
since the sine of 1 mil is -nJW>

D (in mils) «-^^?5^„ 1019 smZ>
' sm 1 mil

76. JO - - 1019 (tan ^-tmE) sin i.

EFFECT OP LATERAL JUMP.

Let / be the lateral jump, in inches, measm*ed on the jump screen,
plus if to the right. Let i be the deflection angle of side jump. Do
not confuse this i with the form factor i, nor with the i of the cant
formula. The other symbols are the same as those u^d in the dis-
cussion of jump in Chapter XI. Notice that here, as in the case of
v^ertical jump, the pivot is taken as half way between the trunnions
and the muzzle.

The horizontal distance, in feet, from the pivot to the screen is

8 8 .

S + K 9^ ^- Su^ o is SO small compared with S, and sec E is so

, 8 ,

nearly unity, that the expresssion may be written jS+h without

material error.

sin I

(^-1)

. , sin i 3438 / 573 /

12(^« + 2J

77. i (in mils) = ^ o . *

Lateral jump is usually not computed, but is included in the
''drift" as tabulated in range tables.

EFFECT OF CROSS WIND.

A 1 m/s cross wind is treated much the same as a 1 m/s range wind
or a 1 m/s vertical wind (see Chap. X) A cross wind is positive, if
blowing from left to right.

A+ 1 m/s cross wind, blowing from time t^ to time T, causes the
trajectory relative to the air to bend abruptly to the left at time t^,

by an angle whose tangent is -,• Therefore the deflection in meters

at time T due to this cause alone will be to a;T — a^t , as 1 to x't : i. e.,

^^7^ to the left.

XII. DEFLECTION FORMULAS. 75

But the trajectory relative to the air will itself be moving to the
right at a rate of 1 m/s for (T—ff^) seconds, or a distance of (T— <q)
meters. Therefore the net deflection, in meters, is:

the values of x and x' being those at time t^.
In mils, assuming the wind to blow throughout the trajectory^

79. 0,=. 1,019(1-^)

DRIFT.

Drift is due to the gyroscopic action of the projectile.

Drift is at present treated as wholly empirical. In range firing
the total angular deviation from the line of fire is measured (posi-
tive if to the right, negative if to the left). From this is subtracted
algebraically the effects of cant and cross wind. The remainder is
the drift (plus lateral jump, of course, which, for range-table pur-
poses, is usually included in the tabulated value of drift).

PROBLEM.

(59) In the firing of problem 58, the right trunnion was 1 inch

-above the left, the distance between the points where the levels were

taken being 3 feet. The shell pierced the jump-screen 9 inches to

the left of the point of bore-sight. The gun is then elevated to 10*^

and attains a range of 6,421 meters. There is a 9.5 m/s cross wind

blowing from left to right. The time of flight is 17.1 seconds. The

shell lands 40 meters right. What is the drift in mils? For the

purposes of this problem do not include the lateral jump in the

<irift.

Questions on Ghafteb XII.

1. If deflection is given in meters, how should it be converted to mils?

2. Why, in the case of cant, is the pivot considered to be at the trunnions?

3. What is the cause of drift?

4. How is drift detenxiined?

-5. What does the range-table value of drift include, besides drift proper?

CHAPTER XIII.
ROTATION OF THE EARTH.'

IN GENERATE

The rotation of the earth has two effects on a projectile in flight:
(a) The higher the projectile goes, the more must its velocity bo^
altered, in order to maintain the same linear velocity relative to the^

f^t

riGii

earth; (6) centrifugal force offsets to some extent the attraction of
gravity.

One of the assumptions on which the computation of a standard
trajectory is based is (see Chap. V): *'2. The earth is motionless.
(The average effect of the rotation of the earth on gravity is included
in the assumed value of jr.) "

The object of the present chapter will be to ascertain the difference
between equations of motion based on the assumption of a motion-^
less earth and those based on the assumption of a rotating earth,,
to separate out from this difference such effects as the standard

> For a nonmathematical treatment of this subject from uiother viewpoint, see ''Physicai bases"'
(Ordnance Textbook 972), pp. 9-11.

ZJU. BOTATION OF THB EABTH. 77

trajectory includes in y©* and to base a rotation correction on the
Temainder.

Inasmuch as most discussions of the rotation of the earth adopt
the convention of rectangular axes (the x axis being tangent to the
earth at the gun), and as the corrections thus derived can be shown
to be equally applicable to the convention of a curved grid (with x
lines concentric and y lines radial), this chapter will develop the cor-
rections on the rectangular basis.

Let us consider a projectile fired from a point whose latitude is I,
in a direction whose azimuth is a, measured clockwise from the south.

Let us consider two sets of rectangular axes, coincident at the
instant the gun is fired. The origin is on the equator at the same
longitude and altitude above sea level as that of the gun; the y
axis is vertical upward; the x axis is horizontal to the east; the z
axis is horizontal to the south. Inasmuch as the orbital motion of
the earth has a negligible effect upon the trajectory, we shall con-
ceive of the axis of rotation of the earth as fixed in space.

Now consider that one set of axes (designated by m) moves with the
earth; and that the other set (designated by/), although coincident
with the moving set when the projectile leaves the gun, thereafter
remains fixed in space, and does not rotate with the earth.

Let t be the time which has elapsed since the gun was fired. Let
R be the radius of the earth. Do not confuse this R with the svmbol
for retardation. Let Q be the angular velocity of rotation of the
earth.^

At time /a, the angle between the two sets of axes isO^A; so, by the
familiar formulas of analytic geometry:

iXai^Xt COS fl^A— (Vf + B) sin O^A
ym-^R^Xtsin Q^A + (^f + R) cos O^a

The subscript m means referred to the moving set of axes. The
subscript/ means referred to the fixed set of axes.
Differentiating equations 80, we get:

81.

x'mm^X^t COS 12^A— J/'ff siu O^a— J/m^— -B^

y'mm==a;'ff sin Qh + y'tt cos Qt^+Xm^Q

2 mm — ^f

The meaning of the double subscript is as follows: For instance,
x'tt means the time derivative of Xf, which derivative is left referred
to the fixed system of axes. If we were to transform x'tt, by formulas
analogous to equations 80, so as to refer to the moving system, we
should get x'fm.

* Since there are 86,164 mean solar seconds in a sidereal day,

0—gg-j«T—. 00007292 radians per second.

COmtSB IN BXTEBIOR BAIXISTICS.

Differentiating equation 81, we get:

82.

.//

y mmm

'"fff COS i^A— J/"fM sin Q^A— 2y'nim^ + XmQ^

:''ut sin QtA+y"ttt cos Q«A + 2x'nimQ+(ym + i?)Q'

•X

The meaning of the triple subscript is as follows: For instance^.
x^'ttt means the time derivative of x'ff, which derivative is left re-
ferred to the fixed system of axes. If we now transform x'^ttt by
formulas analogous to equations 80, so as to refer to the moving sys-
tem of axes, we shall get x"ttm' These formulas are:

83.

.'/

.//

cos IWa— y"fff sin OtA

ffm — X fff „ ^ „,

V"ttm^^"m sin QtA+y"fff

.'/

2 ffm— 2 fff

Substituting from equations 83 in equation 82, we get:

84.

-aj"ffm-2y'„„,Q + XmQ'

= !/ "f f m 4- 2x'„„Q + VmlP + BQ»

*> mmm
y mmm

The symbols with the subscripts w/mm are the components of rela-
tive acceleration; those with the subscripts Jfm are the components
of absolute acceleration; both being referred to the moving set of
axes, which set is that used in computing trajectories.

Equations 84 can now be simplified by dropping the negligible
terms z^S^ and y^S^. Even with a gun firing 100 miles, neither of
these terms could exceed 0.00001 on any part of the trajectory.
Thus:

y "rel « y ' '.be + 2x'rel« + i20'

85.

S"rel=2".b.

Now, regardless of whether we consider the earth as moving or as
motionless, what we are interested in is the motion of the projectile
relative to the earth. If we assume that the earth is motionless, all
terms involving Q in equation 85 would drop out, and we should have:

86.

3C"rel

X"ab.

y''rel«y"ab.

Z"r.x

2f'

abt

If we consider the earth motionless, except for its centripetal
acceleration, the result is to add BQ** to the right side of the ^second
equation of 86. Thus:

87.

* rel — X abs

•//

^<r'/

« rel=2' ab.

XIII. ROTATION OF THE EABTH. 79

Equations 87 represent relative accelerations under the standard
assumptions of trajectory computation. Equations 84 represent
relative accelerations as they actually exist. Therefore the effect
of the rotation of the earth may be found by subtracting equations
87 from equations 85.

88.

Let us now shift the axes until they become the axes used in tra-
jectory computation. First rotate the system northward about the
center of the earth through an angle of I degrees. The origin will
then coincide with the gun. Equations 88 become:

|te"= -20(y' cos Z+2' sin I)
88. j«i/"-+20(x'cosl)

U2"=+20(x'sinZ)

Let us now rotate the x and z axes clockwise around the y axis,
through an angle of (a — 270) degrees. The x axis will now point
in the direction of the line of fire; 2' will equal zero. Equations 89
therefore become:

90.

dx^'= -\-2Qi/ cos I sin a
5y''= —2Qx' cos I sin a
.fe"= -\-2Q{y' cos I cos a + x' sin I)

Now equations 90 were derived on the assumption of a rectangular
system of coordinates, whose x axis is tangent to the earth at the
gun (the 'tangent method '') instead of on the assumption of a sys-
tem whose abscissas are measured along a circle concentric with the
earth and whose ordinates are measured radially from that circle
(the ^'curved method").

It now remains to be shown that equations 90 apply equally to
the curved method. For this purpose equations 90 must be derived
so as to relate to instantaneous axes respectively vertical and hori-
zontal at the point where the projectile is at that instant,* instead
of axes respectively vertical and horizontal at the gun.

In other words, any arbitrary point on the trajectory will be
taken, and equations 90 will be derived with respect to motion at
that point. Then, as that point was any point on the trajectory,
equations 90 .will apply to aU points on the trajectory. That point
will be called the instantaneous point. .

Now revert to page 77, and consider (in place of the two sets of
axes there mentioned) two sets coincident when the projectile reaches

s That Is, the same axes as those of equations 111 in Supplement E.

80 COUBSE IN BXTEBIOB BAUjISTIGS.

the instantaneous point. The origin is on the Equator at the same
longitude and same altitude above sea level as the instantaneous
point. Except as otherwise stated, the development is the same as
the preceding.

tj^ is the time which has elapsed since the projectile was at the
instantaneous point. R is the distance from the center of the earth
to the instantaneous point.

Equations 80 and 84 are developed for a point on the trajectory
very near to the instantaneous point. This near point is then allowed
to approach the instantaneous point as a limit, with the result that
the terms x^^^ and y^^^ become zero and equations 84 become
equations 85.

Equations 86 and 90 evolve as before,* with no change in their
derivation, except that Bi2^ now represents the centripetal accelera-
tion at the instantaneous point instead of at the surface of the earth.
But as gravity is assumed to be constant, regardless of altitude
(which assumption can be shown to cause a^ inappreciable error even
with the most powerful guns), BQ'- can still be considered to be in-
cluded in g.

Equations 90 are thus seen to represent the increments of acceler-
ation due to rotation, and to be correct for either the tangent or
the curved method. In the tangent method they refer to a rec-
tangular coordinate system whose y axis is vertical at the gun. In
the curved method they refer to the directions which are horizontal
and vertical at the particular point on the trajectory.

RANGE EFFECT.

The range effect of rotation of the earth may be found by insert-
ing the values of 5x" and by*' from equations 90 into equation 54
of Chapter X, with the following result:

91. . A Xo = 2fi cos Z sin a I {n/ — px')dt£,

2Q equals 0.00014584.

DEFLECTION EFFECT.

Consider now the deflection effect (D in meters) of an increment
in lateral velocity {8z*) occurring at time <a. Then:

D Jz'

X — x x'
Whence:

4 In the curved method, equations 90 relate to instantaneous cartesian axes which are horizontal and
vertical at the instantaneous location of the projectile. To transform them, so as to relate to the curvilinear
grid of the curved method, we might use equations 120 of supplement £ . But if the increments be regarded
as analogous to inflnitesunals of the first order, the only effect of such a transformation will be to add infini-
tesimals of a higher order, containing ^, which terms can hence be disregarded. Hence equations 90 may
be considered as remaining unchanged.

Xm. ROTATION OF THB EARTH. 81

By difFerentiation^ followed by integration, as in the derivation of
equation 21, we get the following expression for the deflection effect
of an increment in lateral acceleration, occurring throughout the tra-
jectory:

Jo ^
Whence, substituting the value of 6z'' from equations 90:

^^S^ y'dt^ + 20 sin 1 1 (-Y— «) dt/i.
^ Jo

For convenience, the following ''rotation coefficients" have been
adopted:

94.

A = 0.00014584 f^Wp-y'v) dt^.
5 = 0.00014584 P(X-x) dtA.

(7=0.00014584/ -—f-y'dtt,.

Jo *

Note that B is not the B function of atmospheric resistance, and
that Ois not the ballistic coefficient.

Therefore the range and deflection effects of rotation of the earth
become, from equations 91 audi 93:

95.

AXq^A cos I cos a.
Dq^B sin 1+ (7 cos {cos a.

A close approximation to the three rotation coefficients is the value
which would obtain in a vacuum,' namely:

A=OXT(cot«— g tan tp).

B^QXT.

C^^QXTtaiiip.

PROBLEMS.

(60) Prove that 3;mQ'< 0.00001, as stated on page 78.

(61) Derive equations 89 from equations 88.

(62) Derive equations 90 from equations 89.

* See "Physical Bases " (Ordnance Textbook 972), p. 11.
24647—21 6

82 G0I7BSE IN EXTERIOB BAIJJSTIOS.

QuBsnoNs ON Chaftsr XIII.

1. What are the two effects of rotation of the earth?

2. For which of these need the range be oonrected?

3. Do the formulas derived in this chapter apply to the "tangent method" or tO'
the "curved method'^ of trajectory computation?

4. What acceleration is used in the computation of a standard trajectory?

5. What acceleration should be used to account for all the effects of rotation of the
earth?

6. Explain the ratio given under "Deflection Effect."

7. Explain the figure 0.00014584 in equations 94.

8. Firing in vacuo, "what value of ^ would cause the range effect of rotation'oFtho
earth to be zero?

CHAPTER XIV.

COMPTJTATION OF DIFFERENTIAL CORRECTIONS.

The computation of differential corrections is based upon the fol-
lowing equations, derived in Chapter IX.

M"-2A£'yV+[^-%'-?J/' {^^y] /+2 Ex'
' = /%".

96.

'die.

P=- (x'+%V+m')

The method of procedure is to int^rate ii" ahead to get an approxi-
mate value of M^ integrate the approximate m^ to get an approximate
value Hy and then get an approximate value of it," by substituting the
approximate values of /x and /x'^ in the first above formula. Then
integrate ii" to get /x'> and m' to get m; continuing the process of suc-
cessive approximations for any one line imtil the values check.
Thus we employ numerical integration and successive approxima-
tions in a manner very similar to the method of computing a standard
trajectory.

The data is taken from a trajectory computed by the methods of
Chapter VIII. The computing form is made by taking a sheet of
paper about the size of the trajectory sheet of the trajectory compu-
tations, having horizontal blue lines on^-f ourth inch apart. Rule a
vertical Ime H inches from the left-hand margin, then one one-half
inch from the first, then one one-half inch from that, then every
inch or three-fourths inch across the page.

Two special tables are used, being entered with j^ as an ailment.

These tables are:

Table of f {v) = 10* (^ +y ^)
Table of (n-2)

C0UB8E IN EXTEBIOB BALLISTICS.

On the computing sheet, the times run across the pi^ instead of
down it, and run from T to 0, instead of from to T,
The first four columns are filled in as follows:

No.

Next.

Number

of
deelmals.

2
3

4
5
6

7
8

10
11
12

13
14
15
16
17
18

20
21

23
24

25
26

28
29

30
31

13
2

42
10

16
11

3, (24)

27
15

32
60

2
2

2
2
2
2

4
4
4

2
2
2
2
2
2

2
2

1
2

1
1

2
4

1* faDDrox.)

f» ^M^^AV/A.y... ........

t/ ^approx.)

fm ym^^M.\f.Mm.,j ..........

Es/

l(fi. 2h W

f/ ( )

•

9 \ /

(i)=£-y'( )

2ft El/ u

+ (1) l/

1 \*/ " .............

+2 Eoc^

1 " •M^'M ..............

j/^=8um

fm m,^»m^» ............

M=MT- f/dlA

J H

(f)=0.01(MT-AHo).-.-

Deiifl.=(«)X8.310....

4-hi/u

T^'*y M

T^M

niiTTi

sum

nsss

^ '

W"=^2/

xrv. COMPXJTATION OF DIFFEBBNTIAL. COBBECTIONS.

r {?^-Jt)

(5)=l(r*a:^p

(tf)=l(r»y/,r

(7)=(e?-5)

W=JJ{7)<ftA

(P)=0.0001 (X-x)
a

(20)=: j^ (9)dtA ..,

ai)=J(9)

(if)=JJ(jri)<ftA..

a--.v

a.)=(-^

»r,=r-«o-»

(J4)=0.001m

(i^)=1.036(i5)....,

Wy^{ie)-'p

n-2

(i7)=(n-«)^

a 4.

ft !

(ii)=JJ(i7)(ftA...

AXr=0.005 (15)....

No.

Next.

Number

of
deeimftls.

49
50

51
26

2
2

Finals.

86 G0UB8B 127 EXTEBIOB BALUSTICS.

The fourth column above ("number of decimals") need not b©
copied. It is given above merely to indicate the correct number of
decimals to which to carry the computations.

Put the total time of flight ( T) at the head of the fourth column,
and at the head of the succeeding columns put all preceding whole
values of t, from T back to zero.

Fill in lines 4, 5, 6, 7, 42, 50, 26, 16, 11, and 31 with data from th©
trajectory computations, and lines 60 and 9 with data from the
special tables. It is optional whether or not to fill in on the trajeo-

tory computation sheets themselves the values oi E, X— x, —,j etc.

Line 8 is computed from line 7, using + 0.0001036 as the value of
Ti. Line 10 is computed from lines 5 and 9. Line 12, from lines
10 and 11,

The numbers in the colunm labeled "Next" show which line to
proceed to, after completing the line in which the number occurs.
In the absence of any number in the "Next" column, proceed to
the next line below. These numbers have no relation to the italicized
numbers in parenthesis in the left-hand column.

The work, up to line 12, has been line by line across the page.
But from now on (imtil line 23 is completed in all columns), we pro-
ceed to solve each column by numerical integration and successive
approximations.

The procedure is to start with values for /* and /x', obtained from
the following formulas. For time T, the following are precise:

1 x't

m'-O
m"-0.
For the next two columns the following are approximate:

.M"-2y£:T(r-<)p-'

y T

The values of /* are useful merely for a check.
Thereafter the approximate values of m cmd ti' (to be entered in
lines 2 and 3) may be found by integrating /x" ahead by formula 35:

/dt = 2i/t+t(*t+^t+dt . . . )

XIV. COMPUTATION OF DIFrEBENTIAL CORBECTIONS. 87

to get the increment of fi^; and then integrating n' by formula 34:

to get the increment of /x. /* is positive throughout; m' is negative
throughout.

Although we are integrating from time T toward time zero, the
student should proceed exactly as in the method laid down in Chap-
ter VIII. Wherever it is necessary to change signs in integrating,

that fact is indicated by a minus before the | in the left-hand column

in lines 19 and 22. The first, second, etc., differences will be formed,
proceeding /r(>m T toward zero*; but the time interval (i) will be
treated as positive. For a mathematical explanation of all this, see
supplement B.

Note that the time interval between the first and second column
is different from the uniform time interval of succeeding columns.
The first difference obtained by differencing the first and second
columns is used only in integrating for values for the second colunm.
There should be no second difference in the third column, no third
difference in the fourth column, etc.

The procedure, therefore, is as follows: Insert the precise values
for time Tin lines 2, 3, 19, and 22 of the T column. To get the first
precise value of n, it wiU be necessary to use straight multipli-
cation and division, preferably on a calculating machine, as
logarithms will not give results precise to enough places. Insert the
approximate values in lines 2 and 3 of the next two time colunms.
Use these values in lines 13 and 14, and thus get three values of m"
in line 16. Form the first differences of /i", and integrate for the
increment of m'> using formula 34 above. Similarly integrate for
the increment of n. The values thus obtained will usually check
so closely with the approximate values, that m" will not have to
be recalculated.

Start each succeeding column by integrating fi" ahead to ^et
the approximate increment of /x'. Add this algebraically to the last
precise value of n', to get the new approximate /x'« Integrate this
approximate / to get the approximate jx. Use these values in lines
13, 14, and 16, and proceed as before.

After completing the fourth or fifth column of computations, it may
he well, though not essential, to extrapolate back for values of fx", /*',
.and their first, second, third, etc., differences for a time equial to the
next whole number larger than T, i. e., one whole second larger than
the time recorded at the head of the second colunm of computations.

- — -■---■■- _ — — _ - - — _ ■ - _

1 That is, 04 equaisfi—fi. See the footnote on p. 30.

88 COUBSE IN EXTERIOB BALLISTICS.

Starting from this point we can now perform our integrations, with
a constant time interval and a full set of a, ft, e, etc., for use in the
integration. This results in smoothing out the initial steps and in
giving them a greater precision.

When ^' and m have been computed clear across the page, we are in
a position to compute the differential corrections. All of the following
work is done a line at a time clear across the page.

First comes the effect of an increase of 100 gm in the weight of a
cubic meter of air (using lines 24 and 25) ; from formula 66 :

AJCh^ 0.0831 (mt-mO.

The value for time zero is the range-table value, the other values being
used as a basis for weighting-factor ciurves; as will be explained later
(see CShap. XV) .

Then comes (in lines 26-34) the computation of the auxiUary
variables p and v.

Then come (in lines 35-49) the three rotation coefficients, formula 94 :

^ = 0.00014584 J^ Wp-y'v) dt^
B« 0.00014584 J*^ (X-x) dt^
(7« 0.00014584 r{^:Z^\y' dt,

which enter into the formulas for the range and deflection effects of
the rotation of the earth. A, By and C are respectively obtained by
multiplying by 1.4584 the values, for time zero, in lines 40, 44, and 48.
Then come (in lines 50-59) the effect on deflection of a cross wind,
and the effect on range of a range wind or a vertical wind, 1 m/s
being taken as the unit wind; formulas 78, 69, and 71 :

AXwy = — P<o + A I M^^A

The value for time zero is the range^table value, the other values
beng used as a basis for weighting-factor curves, as will be explained
later. (See Chap. XV).

Then comes (lines 60-66) the range effect of the change in elasticity
due to a 1 per cent increase in temperature; formula 64:

A-X'r--0.005r (n-2)^'dt,

XIV. COMPUTATION OF DIFFERENTIAL COREECTIONS. 89

The value for time zero is the range-table value, the other values
being used as a basis for weighting-factor curves, as will be explained
later. (See Chap. XV.)

This completes the calculations of the columns. We now proceed
to certain formulas which are based merely upon initial or terminal
data; formulas 53, 52 and 57:

AXv «= [v COS ^ -fp sin 4] t^o
AJCo^O.I (mo-^Mt)

These are, respectively, the effects of a 1-mil increase ii^ </>, a 1 m/s
increase in F, and a 10 per cent increase in C.
They can be checked by the following (see problems 54 and 53) :

r ^ • ^1 r P ,M'sin0"|

^'-['-w^].

&-} for time zero being taken from line 33.

The computation sheet is completed by the following siunmary of
effects:

1 m/s change in F.

1 mil change in 4>,

Rear wind, 1 m/s.

Vertical wind, 1 m/s.

Ooss wind, 1 m/s.

100 gm/m' increase in density.

10 per cent increase in 0.

1 per cent increase in absolute temperature.

Rotation A,

Rotation B,

Rotation 0.

PROBLEM.

(63) Compute the differential corrections complete from the fol-
lowing data: F= 579.1 m/s; 0=2.3; = 5^; T = 8.824 sec.

COTTBSE IK EXTEBIOB BAIXISTICS.

•

576.9

50.5

0. 1075

3354

520.2

36.2

0.0994

2719

548

473.0

23.5

0. 0910

2242

1044

433.7

12.2

0.0823

1882

1496

401.2

1.8

y ................

0. 0732

1609

100

1913

8.824

374.7

-7.8

0. 0636

1404

2301

353.3
-16.9
0. 0540

1251

2664

336.2
-25.7
0.0453

1137

3009

322.4
-34.2
0. 0388

1051

3338

312.9

-41.1

0.0344

t;2

996.1

100

3599

Questions on Chapter XIV.

1. Why are the times, tabulated in the first row of the computing form, designated
by to instead of tA or t? Why by t in problem 63?

2. Why the minus sign in the terminal value of m? Compare equations 46 of Chap-
ter IX.

3. Why is no first difference formed between the first two columns of computa-
tions?

4. How 19 the first part of the computation smoothed out?

5. The values from what columns are the range-table values?

6. For what are the values from the other columns used?

7. What difference in methods or algebraic signs, from those of Chapter VIII, is
used to compensate for the fact that in the present chapter we are integrating from T
to earlier times?

8. Suppose a tabular function for times 4 seconds, 3 seconds, and 2 seconds, i. e ,
/4,/s, and/2. What is the value of a,? What would this have been in the notation of
Chapter VIII?

CHAPTER XV.
WEIGHTING FACTOR&

In considering how weighting factors are derived from the compu-
tations described in the preceding chapter, let ns take as an example
the weighting factors for range wind.

The computation sheet furnishes, among other data, the effect of a
1 m/s wind on the range. This is entered in the range table, so that
a battery commander, by multiplying this by the nimiber of meters
per second in the range component of the wind, can calculate the
wind correction to apply to his map range.

But this step by the battery commander is based on the assump-
tion of a uniform wind at all altitudes, which is a condition that sel-
dom, if ever, exists. The wind constantly changes both its velocity
and direction from one altitude to another. Velocity usually in-
creases with altitude. Accordingly, some means must be devised for
calculating the velocity and azimuth of a purely -fictitious uniform
wind which woidd have an effect on range equal to the combined
effect of all the actual winds met by the projectile in its flight. This
fictitious wind is used by the battery commander just as though it
were the actual wind. It is called the "ballistic wind." The Brit-
ish call it the "equivalent uniform wind," which is a very apt name.

In the field, the meteorological service measures the average direc-
tion and velocity of the actual wind for successive strata of 250
meters each.

Our problem is now to determiae, for any given trajectory, what
weight is to be given to the wind of each stratum in making up the
ballistic wind. This is done by determining what proportional part
each stratum plays in making up the total correction for a 1 m/s wind
on the computation sheet of the preceding chapter. The procedure
is as follows:

From the trajectory sheet of the original trajectory computations,
find the time corresponding to j/' = 0, and the maximum ordinate (i. e.,
the value of y corresponding to this time,) by methods analogous to
those used in getting the terminal values:^

(a^- :^

^ A^+1, ■ (A< + l) (A^ + 2) .

97. I ^+-Tr^-^ 31 "+•••

,, .. , Mr, , ^<(^^ + ^) X , A^(A< + l)(A< + 2)

> The Technical Staff practice is to use the method for slide rule given in footnote, Chapter Vni, for
finding terminal conditions, adapting it to the present conditions.

92 COUBSE IN EXTERIOH BAIXISTICS.

taking the values of y' and its a, i, c, etc., and of y and its a, h, c, etc-r
from the line whose y' is nearest to zero. In the first equation, use
the a, h, c, etc., of y' and solve by successive approximations, taking

for a first approximation A(-=-Jl. Ja the second equation, use

the a, h, c, etc., of y. The result is i/., the maximum ordinate.

si mz

:::

''^;

RG 12
Against t tabulate y from the trajectory sheet and T—t^ — vt^ from
line 53 of the differential computation ^eet, and calculate y imd

T-h-

-^, as follows:

T-t.-.^

T-U

''-^-'-

^-'o

This tabulation gives for each value of to'. The altitude of the pro-
jectile; the range effect of a 1 m/s wind blowing from time U to time
T; the ratio of the altitude to the maximum ordinate; and the ratio
of the aforementioned wind effect to that of a 1 m/s wind blowing
throughout the entire flight of the projectile.

XV. WEIGHTIKG FACTOBS.

rp M

Now plot, on a small sheet of coordinate paper, — ttt;^ — - against

— , and connect the points by a smooth curve. (See fig. 12.) This

curve will run from the point (1, 0), representing time zero, to the
origin, representing time T; and will be tangent at its summit to the

line^ = l.

Let AB be drawn parallel to the axis of abscissas, a distance of any

— above it. Then EB represeiits the proportional effect of a 1 m/s

wind blowing from the time the projectile first reaches altitude y,

until the point of fall. EA represents the proportional effect of a 1 m/s

wind blowing from the time ,q

the projectile reaches altitude

-y in its descent until the point ^

of fall. Therefore AB repre- «

sents the proportional effect

of a 1 m/s wind blowing above

the altitude y.

On a second sheet of coordi-
nate paper, plot AB (meas-
ured to the left from the line

of unit abscissas) against ^^

<Seefig.l3.) Then FJ5 equals
1—AB, equals the propor-
tional effect of a 1 m/s wind
blowing below altitude y.

The proportional effect of
a wind blowing throughout any stratum is the difference between
the FB corresponding to the top y of the stratum, and the FB cor-
responding to the bottom y of the stratum.

Thus, to get the weighting factors for the trajectory in question,

mark off on the curve like that of figure 13, points whose — are respec-

.3
2
.1

—

p —

X^"

•••

JIIU

""' "

""■*

^" ~

xsss n

•

■

r-~

' 1

r—

£1

\f\\

JIEfl

^Si

(lUki

Win 1-

H a

> 'K

•1.

1
«

■

1.0

FIG 13

tively

250 500 750

» etc. The difference in abscissas between the

y. yB ys Vs ^

^rst and second points is the weighting factor for the first stratum;
the difference in abscissas between the second and third points is the
weighting factor for the second stratum, etc. As a check, the sum
of all the weighting factors should be unity.

Of course, it would be possible to calculate the weighting factors
directly from the curve like that of figure 12, but it is more convenient
to have the weighting-factor curves in the form of figure 13, as it is

94 GOUBSE IN EXTERIOR BAIXISTICS.

usual to plot on the same sheet the curves for various angles of
departure of the same gun.

It should be noted that, for any given trajectory, the weighting
factors for making up a ballistic wind to use for range corrections are
quite different from the weighting factors for making up a ballistic
wind to use in deflection corrections. Therefore, according to present
methods, there are for each combination of (7, V, and </>, two ballistic
winds: (a) The range ballistic wind, whose range component is used
as a basis for range corrections; and (b) the cross ballistic wind,
whose lateral component is used as a basis for deflection corrections.

The foregoing method for getting weighting factors for range bal-
listic wind is equally applicable to cross ballistic wind and ballistic
density, the last requiring slight modifications of a fairly obvious
sort in referring changes aloft to equivalent changes at the ground.
The treatment of ballistic elasticity has not yet been decided upon.

For each combination of G and F, a chart can be made up showing
(for instance) the weighting factor curves for range ballistic wind for
four or five values of <t>. The curves for other values of can be inter-
polated, if only we have a curve for = 0. But of course no trajectory
can be computed for <^=«0. Nevertheless, there has been deduced
the equation of the limiting curve •which the range wind weighting-
factor curve approaches as 4> approaches zero. The equation is:

in which p is the abscissa and Ic the ordinate of any point on the
curve; and n is the G&vre n, to be obtained from the table of (n— 2),

entered with r^, using the standard muzzle velocity.

The equation of the limiting curve for density weighting factors is:

99^ l-p=|(l-i)v«+i(i-i)V.

The equation of the limiting curve for cross wind weighting factors is •

100. i-p^ii'-Tcyi'

The Umiting curve for elasticity weighting factors is the same as
that for density; but, owing to the fact that elasticity weighting
factors become infinite in certain cases, the use of such weighting
factors is not to be reconmiended.

It will be observed that in the case of density, elasticity, and
cross wind, the Umiting curve is independent of muzzle velocity and
baUistic coeflB.cient. In the case of range wind the Umiting curve
depends on the muzzle velocity, but not on the baUistic coefficient.

The weighting-factor curves for ^ = can be plotted on the weight-
ing-factor charts with very Uttle extra work, and wiU assist materiaUy
in the interpolation of new curves among those computed.

XV. WEIGHTING FACTOBS. 95

By comparing a large collection of weighting-factor curves at
Aberdeen, three mean wind weighting-factor curves have been
deduced, whose equations are, respectively:

1-2> = 1.11 (l-?fc)«-0.11 (l-Jcy

101. l-p = 0,74 (l-]fc)H+0.26 (l-fc)»

l-p = 0.36 (l-Xr)« + 0.64 (l-Xr)*

For any given battery, one of these curves can be chosen, which
will give a sufficiently approximate baUistic wind for^both range and
deflection corrections for actual field service.

Similarly the following single density weighting-factor curve has
bedn deduced:

102. l-p = 0.48 (l-i)« + 0.52 {1-Jcyi*

Prior to the World Wai* a single curve, known as the '' time curve"
or "vacuum curve" was used for all weighting-factor purposes.
These names are due to the fact that this curve weights the various
strata of atmosphere in proportion to the time the projectile would
spend in each, if the trajectory occurred in vacuo. Its equation was
the same as that of the limiting curve for cross wind (equation 100).

The present indications are that the second equation under 101
will be adopted by both branches of artillery^ for all wind weighting-
factor purposes; and equation 102 for all density- weigh ting factor
purposes.

PROBLEMS.

(64) The maximum ordinate of problem 63 was y,— 97.9. Con-
struct the range-wind and the cross-wind weighting factor curves on
the same sheet of cross-section paper.

(65) Plot equations 102 and 100 on a single sheet for comparison
of the old and the modem mean density weighting-factor curves.

(66) Plot equations 101 and 100 on a single sheet for comparison
of the old and the modem mean wind weighting-factor curves.

Questions on Chapter XV.

1. Define "ballistic wind."

2. What is the British term for ** ballistic wind? '»

3. What are ** weighting factors? "

4. Explain the two ballistic winds used in present-day methods? Are these the
components of a single ballistic wind?

6, As no trajectory can be computed for = 0, how are the weighting factors
obtained for this angle of departure?

6. Of what use are the curves for =« 0?

7. Are weighting factor curves used to determine ballistic elasticity?

8. Define "ballistic density."

9. What is meant bv the "time curve"?

s As a result of experiments at Fort Monroe, the Coast Artillery has adopted the tecond of equations
101 for all wind purposes.

CHAPTER XVI.
CONSTRUCTION OF A RANGE TABLE.

A range firing consists of a number of roxmds, usually 10 to 20,
fired at each of several elevations, say, at 5°, 15®, 25°, 35®, and 45^,
The exact location of each point of splash is plotted by means of
intersecting azimuths from at least four observation towers.

Throughout the firing the meteorological conditions are observed
from time to time by aeroplane, pilot balloon, etc. At the gun a
detailed record is kept of the time of firing, quadrant elevation, and
ot all variations from standard, such as weight ot projectile, cant of
trunnions, etc. Tiine of flight is taken by stop watch as a check on
the computations later to be made and as a basis for the rough deter-
mination of maximimi ordinate (j/b = 4.05 7^).

Separate roxmds are usually fired through jump and chronograph
screens, as a basis for determining jimip and muzzle velocity.

Given the range-firing records for a specified gun and projectile,
with prescribed values of weight of projectile and muzzle velocity,
to construct a range table for this gun and projectile, the procedure
is as follows:

Divide the total roimds fired into groups, each having identical
ranges (except tor sUght differences due to dispersion) because of
having been fired at the same elevation and on the same day.

Compute, for each group, the mean values of all measured quanti-
ties. These measured quantities are muzzle velocity, weight of
projectile, observed range, deflection, right wheel above left, time of
flight, etc.

With average values of the atmospheric conditions, and with
weighting factors from a similar range table, compute a tentative
ballistic range wind, cross wind, temperature, and density. If no
similar range table is available, use the mean weighting-factor curves
of Chapter XV.

Compute, for each group, the angle of departure, by correcting the
quadrant elevation for any individual error in the quadrant, for any
inclination between quadrant seat and axis of the bore (determined
by applying a clinometer to the gun) for jump and for height of site.
(See Chap. XI.)

Unless velocities were taken on the range rounds themselves, esti-
mate the mean muzzle velocity of each range group, as follows: Let
bp be obtained by substracting from the mean projectile weight of
the group in question, the mean projectile weight of the velocity
96

XVI, CONSTRUCTION OF A EANGB TABLE. 97

rounds fired on the same day. Then the estimated velocity of the
range group will be the algebraic sum of 8Y and the mean velocity of
these velocity rounds, 5 F being obtained from the equation:

dV 8p

^ P

Y^n-^^

See the explanation preceding formula 58.

Estimate the ballistic coefficient for each group. This may be
done in a niunber of ways. The observed ranges should first be
roughly corrected for nonstandard conditions^ using a similar range
table; or the Ingalls tables, or Alger's charts (which are a graphic
representation of the Ingalls correction formulas), or French charts,
or the A. L. V. F. tables, or the GAvre tables of September 15, 1917.
The last-named two are French tables. An Americanization of the
A. L. V. F. tables has been published by the Ordnance Department
(War Department, Document No. 983, Conjidential), and an Am«i-
canization of the GAvre tables by the Coast Artillery Board (mimeo-
graphed — title: "Artillery Ballistic Tables").

Then with <^, F, and the corrected X as arguments, enter ''Ingalls'
Ballistic Tables'' (printed by War Department, 1918), or a set ot
French charts, or a set of Alger's charts (Journal of the U. S. Artil-
lery, Dec. 1919, p. 585), and take out 0. It should be noted that
the O obtained by either of these methods is not the (7 desired. The
C of the Ingalls tables and the Alger charts is the Siacci C, so-called:

The O of trajectory computations in modem bidlistic methods is the
normal 0, so-called:

m. c«=5.

The coefficient of form, (i) of the two O^a is also slightly dif-
ferent. But a tentative value for Cn can be obtained by extracting
j8 and/a* from (?». The use of French tables or charts is, however,
much better.^

105.

logiu(^)= -0.00012 T^.

1 -7 isthemeaavahteof ffforthetrajeetwy. Thte ia awrnnad to be the same as tha gat an altitude

two-thirds the marimnTn ordinate. The Ingalls formula for maximum ordinate in terms of time of
^ghtlsy«>»4.057^. ThereCoce:

logio(j-) -logioff— 0.000045 (^)— 0.00003 y.—0.0(X>1216 T*.

24647—21 7

98 COURSE IK EXTBRIOB BAIZ.ISTICS.

The French C and the nonnal C are bound together by the follow-
ing approximate relation:

1 OR flog^Cr +log„C« = 7.0570 - 10

*"*• ICp . (7« = 0.001 140.

The Americanized French tables give ft direct, instead of C-g-
Flot the approximate values of C thus found against ^, and draw a
smooth curva For each value of * at which firings were made, take
the smoothed-out value
of C and the standard F
and compute a traiectoiy
by die methodslaid down
in Chapter VIII.

Compute the difTeren-
tial corrections by the
methods of Chapter XIV
for each of the computed
trajectories.

The variations from
standard for which the
differential corrections
are to be computed are as
follows;

1 m/s change in

muzzle velocity.

1 per cent change

_._ .. in atmospheric

density.

1 per cent change in absolute temperature (as affecting ela&-

ticity).
1 m/s range wind,
1 mil change in elevation.
1 per cent change in weight of projectile.*
1 m/s cross wind.
From these results, compute the changes in range for the variation
of actual conditions from standard (atmosphere, muzzle velocity,
weight of projectile, and elevation) at the time of firing, and correct
the observed ranges to standard ranges (i. e., to the ranges which
the given i^ and V, and the estimated C would have produced under
standard conditions).

■ Only the But umt ol lonnula SS sboiild be usei in gatting tbg standard range from tbe oinerred range-
In getting tlie eitinutaj velocity <i the range rounda Irom tie observed veloti^ roundB, nee -p-n ^ as
before. [Bee the explanation vUch precedes [ormnla 68.)

XV]. CONSTEUCTION l)F A BANGE TABLE. 99

Compare these corrected ranges with the ranges obtained from the
trajectory computations. If the differences are considwahle, say,
greater than 5 per cent, correct the values of the ballistic coefficients
at which the trajectories were computed and recompute the trajec-
toriee and the differential corrections for the new values of C. Then
repeat the work indicated in the precedii^ paragraph.

The range-elevation curve should now be constructed. Correct
bach quadrant elevation (i. e., observed elevation) by adding the
angle of site and the corrections for the quadrant and quadrant seat.
We do not here correct for jump, as what is here wanted is the angle
of elevation rather than the angle of departure. Plot the corrected
ranges against the corresponding elevations as thus obtained. (See
fig. 14.)

RG. 15

Certain elements of the trajectory and certain differential correc-
tions should now be plotted against the angle of elevation correspond-
ing to the angle of departure from which they were computed. (See
fig. 15.) On one sheet of cross-section paper should be plotted —

Slope of fall.

Angle of fall (w).

Timeof flight (r).'

Terminal velocity (vt).

Maximum ordinate (y,).

100

OOUBSE IN BZTBBIOB BALLISTICS.

On one sheet should be plotted (See Fig. 16) the rai^e changes due to —
1 m/s mcrease in muzzle velocity.
1 per cent decrease in atmospheric density.
1 per c(mt increase in absolute temperature (only ae affecting

elasticity).
1 m/s following wind.

5 V IS2OZ5 3OJ540 45

<IN6t.E OF ELEVATIOH IN OEGREtS

FIG 16

MILS

5 ■ ' S

!: ::;::::: : -

„ ^^»

- - :;:::::_ : _

» [l ^.

fe- u THL+-J

s° I I ;

"-. "

: '-, i

i- t ;

^ i

" _ u^ :_:

: _._ '-,^

Sj - . i - . --

,^^

1^4 _ ■ . :::"

,^^_

XVI. GONSTBUGTION OF A RAJNGE TABLE.

101

1 mil increase in elevation.

1 per cent increase in weight of projectile.*
On one sheet should be plotted (see fig. 17) the deflection effect,
in mils, of —

1 m/s cross wind.

Drift (including lateral jump) .
Compute the probable error for range and deflection. This is
obtained by multiplying the mean deviation by 0.S45. UsuaUy 20 or
more rounds are fired at each of several elevations for the determina-
tion of dispersion alone. The dispersion of the range groups is also
considered if these contain enough rounds. Plot the 50 per cent zones
against elevations. The 50 per cent zone is twice the probable error.

Ip .

— 1

1 —

^^MB

— ■

t»—

■~1

3i — <

1*1

iTl

?■

fT-

L ,rf

kX^

l»^

rr'

'm.^

■"" ■

—

._.

• —

—

. _.

- ■

=■

t—

—

. .

\&Ll

r :l

"^T

r r

^ 1

>^<

b ^ 2

FIG. 18

Any additional data which may be needed should be treated in the
same way.

All curves should be smoothed out. Practically no smoothing
should be necessary, except in the case of the drift curve and the
probable error curves.

From these curves take the necessary data for the construction,
<rf the range table.

Typical curves are shown in figures 14 to 18, inclusive, which are*
based on the computation of a range ' table for the 75 mm gun, firing
a Mark IV projectile at 1,900 f/s.

< The whole of f ormtila 58 should be used here.

102

COURSE IN EXTERIOR BALLISTICS.

PROBLEM.

(67) Construct the necessary curves from the following data;

155 mm G. P. F. gun firing 94.7-pound shell at 2,410 //« velocity.

IJ umber of rounds considered

Clinometer elevation

Range (meters)

Deflection (mils)

Time of flight (seconds.)

Weight of projectile (pounds)

Mean deviation in range (meters). .
Mean deviation in deflection (mils)

Ballistic range wind (m/s)

Ballistic cross wind (m/s)

Ballistic density

Ballistic temperature (°F)

Right wheel above left (meters) . . .
Distance between wheels (meters) .

Angle of bore sight

Height of trunnions above mean

low water (meters)

Height of tide (meters)

Vertical jump (minutes)

"Weight of projectile for velocity

rounds (pounds)

Velocity of velocity rounds (f/s)

Date.

June 9.

25
6° A'
6,477

5.9

13.23

93.48

0.5
+2.3
-fl.8
0.978

70°

0. 0033

0.503

6° 4'

5.94

0.54

-2.0

93.33
2,391

June 9.

16° 35^

11, 288

10.7

29.85

93.67

0.5

+3.4

+2.5

0.982

70^

-0. 0027

0.503

16° 35^

5.94

0.40

-2.0

93.33
2,391

June 9.

16° 35^

11, 453

8.1

30.06

93.53

0.6

+3.7

+0.6

0.981

70°

-0. 0027

0.503

16° 35^

5.94

0.40

-2.0

93.33
2,393

June 10.

25° y

14, 142

6.3

41.89

93.52

107

0.7

-5.4

-2.9

0.972

65°

-0.0003

0. 503

25° 5'

5.94

0.61

- 2.

93.35
2,401

June 14. ,

35° ^

16, 564

7. '4

55.0

93. 37

0.6

-8.8

-9.0

0.965

71*

-0.0027,

0.503^

35^ ^\

i>.'94
■ 0.43
-M

95.64
2,410

QuEsnoNB ON Chapter XVI.

1. Give a very general outline of the steps in the construction of a range table. *

2. How may the ballistic coefficient be estimated?

3. Why do you suppose y, equals 4.05 T*, instead of 4.02 2^?

4. In constructing the range-elevation curve, why is not the observed elevation^
corrected for jump?

5. To convert the observed range to the standard range, should the results of solving
the formulas of Chapter X be added or subtracted? Why?

6. What curves are likely to need smoothing out, and why?

Supplement A.

TRAJECTOHY COMPUTiTION BY THE TANGENT RECIPROCAL METHOD.

The tangent reciprocal method of computing trajectories is a
variant of the rectangular method described in Chapter VIII. It
is based on the following equations, involving three auxiliary vari-
ables (Ty a' J and 0-

//.

107.

,f^f

j« J^ « _y_?L (i, e tan B divided by g)
x'g g J Iff ^

, 1 (i. e., minus the reciprocal of the horizontal com-

.ff

Er'

ponent of velocity)

These three variables are "critically varying", i. e., they change
at such rates that inaccuracies produce the least possible effect on
the results. This enables the use of longer tin?e intervals ^ith the
tangent reciprocal method than with the original rectangular method.

For the tangent reciprocal method, change the trajectory sheet as
follows: Use the — Ex' colunm for y. Use the velocity, mean height,
and time colunms, and the first half of the Jy dt column for y\
Use the last half of this colunm for time. Use the y column for <r,
the y' column for <r', and the —Ey'—g colunm for a'\

On the small computing sheet, label the rows as follows (the
numerals show the order in which the rows are used) :

log(?..
logF...
colog C
log-B...
log</..

colog </
log^....

iQg^....

log/..

x^/100,
vVlOO..

Order.

9
ConBtant.
10

1
11

2
Constant.

103.

104 GOTJBSE IN EXTEBIOB BALXJSTICS.

The logarithms are all denary.

To start the computation, enter on the trajectory sheet following
initial values:

X =

x' =

Fcos ^

y =

!/' =

Fsin ^

<r' =

a =

y'<r'

Enter on the small sheet log H=0, for t = 0; colog C (constant

the table of -squares; add them to get yoo'* ^^^ y^^th. this as an arga-*

ment, get log from the 6 Table.

Add log 6y log Hf and colog C to get log E. Add log E to log <r^
to get log <r ". Enter <r" on the trajectory sheet. Note that initially a
is plus, and <r' and a" are minus. This completes the computation
for^=0.

To start a new line, integrate a'' ahead to get the increment of a\
Integrate a' to get the increment of a.

Turn to the small sheet. Set down log <r^ From this, get colog a\
Set down log a. Add colog a\ log g and log a to get log y\

Enter the table of logs and squares with log x' (same as colog a')

^.o.v.«.^^.„..dt.keo«t.-„.,^.nd.^. Add ^.„.

two to get r^* With this enter the table and take out log 0.

Integrate y' to get the increment of y; and, with y as an argument,
get log H from the formula

log H= (10-0.0000452/) - 10.

Add log 6, log H and colog C, to get log E. Add log E and log a\
to get log a".

Integrate a^' to get the increment of o-', and <r' to get the increment
of a. If these new values check sufficiently close, the work on tiiat
line is completed.

Proceed in the same way with each successive line.

Supplement B.

EXPLANATION OF THE SIGNS IN THE COMPUTATION OP DIFPERENTIAl

CORRECTIONS.

Whatever may be their physical significance, all of the int^rations
of Chapter XIV are, mathematically speaking, performed from a
later *to an earlier time.

In integrating from a later to an earlier time, the first differences
of the integral must receive an algebraic sign opposite to that of the
integrand; in other words, a positive integrand produces an alge-
braically decreasing integral and vice versa. Thus the integral:

I ( ) dtis, requires that a change of sign be made when numerically

( ) dti, does not. But it is bad

psychology to use a notation in which plus requires a change of sign,
and mintis does not; hence, although integrating backward, we shall

use the symb 1 I in place of the symbol I . Then, since

Jt» t/T

( ) dt^^- I ( ) d«A, and

Jr»t» /»T

( )dtt,^+ \ ( )dtA,
T •/to

a minus will meui to change signs, and a plus will mean not to
change.

Chapter XIV employs two sorts of integrals. The integrations to
get pl' and /i are of one sort. Since y/ is the derivative of /i then, m
is the indefinite integral of /*'. That is —

Mto= I iJ-'dtA.
But the values of fjk are fixed by being known at time T, Hence:

in line 22 of the computation sheet.

This equation means that p, is the indefinite integral of M^* that its
value is known at time T; that to find its value for an earlier time,
one int^rates from T to this earlier time; and that one changes sign
in int^rating. Similarly:

/t. pu PT

n''dtA=ii\^ I ix''dtA^ix\- I /i'^d^A.
•/T •/to

The other integrals of Chapter XIV are of a different sort, namely
deiiniU integrals from time t^ to time T, representing the effects, at

105

106

OOUBSE IN EXTEBIOB BAUJSTIGS.

time T, of causes starting at time t^. Thus, in line 40 of the compu-
tation sheet:

(8) = + fV) dt^.

Logically we ought to integrate from each t^ to T, proceeding
from right to left, as follows:

18.700

0.08
-0.22

0.38

1.20

1.77

2.44
Etc.

(7)

0.30
-0.22

0.62
-0.10

0.62
-0.11

0.73

y*^.*.......... .........

(«)-+J[^(r)*A

(8)-+J.*'(7)*A

•

W«+J^(7)d<A

W-+J,**(7)rfiA

Etc.

0.03

0.41

1.23

1.80

2.47
Etc.

0.82

1.39

2.06
Etc.

0.57

1.24
Etc.

0.67
Etc.

Etc.

But, if we are willing to give up logical arrangement, in order to
save labor and space, we can obtain the same results as foUows, inte-
grating from left to right:

to.

16.760

0.30
0.22

0.41

.10

(7)

0.08

0.52
0.22

1.23

0.62
0.10

1.80

0.73

0.11

w-+i

•T

(7)dtA

0.03

2.47

In either way, the plus sign in front of the integral signifies that
no change of sign is made in integrating.

The heavy black line between two columns is a convenient device
to indicate a change in time interval.

To recapitulate, although all integrations in Chapter XIV are made

£T
, SO that a
-J
minus sign before the integral shall serve notice that the increment of

the integral is to be given a sign opposite to the sign of the inte-
grand, and so that a plus sign shall serve notice that no such change of
signs is to be made.

Supplement C.

DIMENSIONS OF BALUSTIC SYMBOLS.

In deriving ballistic formulas, it is frequently convenient to test
them by the theory of dimensions. For this purpose we shall employ
ihe following three dimensions: Length (Z), time (T), and mass ( Jf).
Unity will be expressed by 1.

The following are the dimensions of the principal ballistic symbols:

Symbol. Dimensioiis.
xtokdy L

a/yi/y andi» L/T

a;^^/^andsr L/T*

O L/T

H ; M/L*

^ VL

C M/L*

v> M

i 1

•^ '-/^

■"•('+»'-^ y-

E 1/T

M L

m' L/T

Mf' , L/T*

j/andp T

i/andp^ 1

AU trigonometric functions 1

All angles 1

All logarithms 1

All exponents 1

All of the foregoing dimensions are fixed by physical laws except

ihe dimensions of 0^ H, and 0.

There is a wide latitude possible in choosing the dimensions for

OH 1
these three symbols, provided only that the dimensions of -^ are y-

107

108 COUBSB IN BXTEBIOB BAUJSTIGS.

Thus may be treated as a dimensionless constant and H a dimen-
sionless ratio; G will then have then same dimensions as E, which
convention would have much to commend it. Or vO could be
regarded as having the dimensions of its Mayevski equivalent An v^t

i. e., jq^» whence O would have the dimensions ^*^::i-
The dimensions actually adopted for 0, H, and C were arrived

/«X ay

at as follows: was regarded aavB l-h — being treated as a dimen^

sionless ratio (see equations 59 to 64 m CSiap. X). H was given thisr
dimensions of density, and C the dimensions of sectional density..
The following physical laws were thus expressed:

-o

1203.4

The usefulness of the tabulated dimensions is as follows: To test
the dimensional correctness of a formula, substitute for each symbol
its dimensions, disregarding algebraic signs and non-dimensional
numerical coefficients. If the dimensions of all the terms are alike,
it is dimensionally correct. Very often typographical errors and
mistakes in derivation can thus be readily detected.

Supplement D.

ANTUmCEAFT FHIE.

At the time of writing this book the methods of computing dif-
ferential corrections for antiaircraft fire are in such an imsettled
state that it is thought best not to describe them in detail, but merely
to refer to them in a general way.

In Chapter IX we saw that the use of three auxiliary variables,
tiy V, and Pf and an auxiliary constant, X, enabled us to express a
change in Jf, T, or y. in terms of charges in x, y, x', and y' occurring
at any time; the auxiliary variables being lunctions of the time of
the disturbance. The expressions for SXy 8T, and dy^ in that chapter
may be generalized into

108. 5( )=X:&c-hifeM<5y-f j'^x'-hp*!/'

where the parenthesis represents an effect of any given nature occurring
at any given point on the trajectory. Thus dX is the effect on x at
the point of fall, 8T is the effect on t at the point of fall, dy^ is the
•effect on y at the summit. Similarly we might use the same general
form to express any given effect at any other point on the trajectory.
The auxiliary variables here used are not to be confused with the special
•cases considered in Chapters IX, etc.

In the general form given above, 5x, 5j/, 8x\ and 8y^ are arbi-
trary small increments occurring at the time of the cause (which
we shall call i^),

^f Vf ^^ y\ a;", and j/" are functions of the trajectory and of the time
(t) of the point on the trajectory at which their value is taken, whether
liiis be the time of cause (jt = h)f or the time of effect (< = ^«), or some
other time.

X, M, Vf and p are functions of the trajectory, of the nature of the
•effect considered, of the time of cause (<a), and of the time of effect

5 ( ) occurs at time ^o and is a function of the trajectory, of the
nature of the effect considered, and of 8x, by, dx\ dy\ t^y and <«.

Let us now consider some particular trajectory and some particular
•effect, such as change in x. Let us successively consider some fixed
value of te, the time of effect, so that it successively equals 0, 2, 4,

^ . . . r.

Then for each value ot t^:

109. 5xt =X 5x + JiM dy + v 8x'-\-p by'.

For' each value of U, we can get a different set of values of X, n, v,

i&nd p as functions of /a alone.

loe

110 CX)UBSB EST BXTEBIOB BAUJSTIOS.

The entire computation of Chapter XIV is in effect repeated for
each value of te, running each computation backward from ^o=^ to
/o==0, the result being a tabulation of differential corrections, from
which can be determined the effect on ihe z coordinate of a projectile
at any point in its flight, due to a disturbance at any preceding point
or points in its flight.

The terminal values used to start each ot these computations are:

110.

cot e
^ IT

ii! =0
m"-0
V -0

Ip =0.

This is a very tedious performance. Accordingly there have been
devised several alternative methods, each involving variables aux-
iliary to the auxiliary variables, i. e., bearing much the same rela-
tion to the auxiliary variables as the auxiliary variables do to the
elements of the trajectory.

In the latest method, two sets of values for p, and \i' are computed,
each based on the assumption that equation 108 holds true and that
X=»0; one based on the assimiption that %m is zero and m' unity at
the gun, the other on the assimiption that A/i is imity and \i^ zero
at the gun. These four variables, called /x,, /i's, /i4, and \i\^ are func-
tions of the time of cause alone. In this method the integration is
performed forward.

Variable coefficients -K^, Z^, Z^, L^, if,, M^^ N^, iV^, and A are
also computed, these being functions of /x,, fjk^y fjk\, and n\, and of
the time of effect. This A is a quantity, and not an operator.

Attempts are now being made to simphfy this system. Until
this is accompUshed, it is not thought advisable to present the matter
to students in any more detail than is here given.

Supplement E.

DRRIVATION OF TWO EQITATIONS OF CHAPTER VII.

The precise equations of motion of a projectile in the tangent
method (equations 30) or the curved method (equations 31) have
been derived in various blue prints of the Ordnance Department.
A brief skeleton of the derivation is here given.

The equations of motion referred to an instantaneous system of
cartesian axes/ whose origin is the instantaneous position of the
projectile, and which are respectively horizontal and vertical at that
point, are:

Tr.9

tan ^=?7

The tangent method assumes, in place of an infinite number of sets
of instantaneous axes, a single set, namely, the instantaneous set
whose origin is the gun. The only effect of this assumption is to change
the basis of the H. function and the manner in which gravity enters
into the equations. Also, in any system, gravity (jg) ought to be ex-
pressed in terms of the constant surface gravity (^o)-

If ^ is the angle at the center of the earth subtended by the flight
of the projectile up to its arrival at the point XY, if p is the distance
from the center of the earth to the projectile and R the distance
from the center of the earth to the gun, and if gravity varies in-
versely as p*, then:

110 ^ /g'cos^^ /, 2Y \/, ' \

The components of gravity are:

113.

<7^-flr8in^ = <7,(l-^+ • • • ){^- • • • )
j?,=^ cos ^-f^,(^l--g-- • • •)(l- • • •)

Therefore the equations of mot'on are as given in equations 30 of
Chapter VII.

■ ■I '«- II ■ 1 1 |ii III

1 Coordinates x and y are those of the instantaneous system. Coordinates X and Y are t o>e of tne
tangent method. Coordinates x and y are those of the curved method.

Ill

112

GOUBSE IN EXTERIOR BAUJSTIGS.

Also the correct basis of the H function becomes:

114.

^ cos 4/ 2R

The curved method assumes a single system of orthogonal curvi-
linear coordinates; whose x is measured from the gun along a circle
concentric with the earth, and whose y is measured vertically upward
from this circle.

The equations of motion of the instantaneous system are converted
into polar coordinates ^ and p as defined above, and thence into the
orthogonal curvilinear coordinates.

The first conversion is derived as follows: Consider a point Pj on the
trajectory near the instantaneous poiut P, and denote by A ^^ the
angle at the center of the earth subtended by PPi* Then:

115.

35 = p sin A ^

y — p cos A ylf — OP

where is the center of the earth.

Differentiate twice, with respect to time, and then let P^ approach
P. Cos A i^ will approach unity, and sin A ^ will approach zwo, and
we shall have as conversion equations:

116.

S" = 2p'^' + (»(f

=p"-p^'^

The relation between the polar coordinates (^, p) and the coordi-
nates (x, y) of the curved system is:

117.

^ =

[p=y + R

Differentiating twice, with respect to time, we get as conversion
equations:

118,

p'=y'

ft «.//

^'fJ^

E. DEBIVATIOK OF TWO EQUATIONS OF CHAFTEB VII.

118

Substituting from equations 118 and 119 in equations 117:

119.

r-x' +

-'■'My%^

Substituting from equatioas 119 in equations 111, correcting g for
altitude, but not for obliquity, and neglecting y in comparison wiih
22, we get the equations of motion as given in equations 31 of Chapter
VII.

24e47— 21 8

Supplement F,

A DERIVATION OF THEOREM 1.

An inspection of any number of formulas of ordinary differentia-
tion will show that they are homogeneous with respect to the ordi-
nary differential operator d} Therefore, let us assume, as is indeed
the case, that there is no possible form of differentiation whicb^
violates this principle.

Let

«=

=/. {X, y, z,

• •

=-/» (*, y, 2. . .

■=/» («, y, 2, • •

Z-

=/4 («. y, 2. • •

etc.

Then, by the foregoing assumption:

An ^AAX-^-BiY-^- ... (a>

dX^Cdx +Ddy + . . . (b)

dY^Edx +Fdy + . . . (c)

where A, B, Of D, E, F, etc., are undetermined coefficients.
Substitution from equations b, c, etc., in equation a gives:

du-'{AC+BE+ . . .) dx+{AD'\-BF^- . . .) dy-f . . . (d)

In equation a, u was considered as a function of X, Y, Z, etc. In
equation d, u is considered as a function of x, y, s, etc.
Now divide equation a through by dX. Then :

du . . j^dY,
5?"^ + ^5X^ • • •

du
If, in this expression, Y, Z, etc., be regarded as constant, then ^

becomes ^ by definition of the partial derivative, and all the other
derivatives of this equation vanish. Thus A is identified as

OttxTZ • • •

1 This means that, if we treat d as an algebraic quantity, Instoad of as an operator, each term will contain
i to the same power as any other tenn.

114

F. A DERIVATION OF THEOP^M 1. 115

Similarly:

5tlxYZ . . .

dY~

*' dx

P^XfU . . .
ET ^ TfU , . .

^ di

Now divide equation d through by dx, and r^ard y, z, etc., as con-
stant. The equation thus becomes :

Substitute the identified equivalents of A, O, B, E, etc.
Then:

jy^m . . . _ dtixYz . . . dXxfz . . .j^duxYz . . . c^Yxfu ... . n IP n

— ^ bX di— +~dT- bx +• • • <2-^-^'

Supplement G.

NEW METHODS OF TRAJECTORY COMPUTATION.

Since this book went to press, a new method of trajectory com-
putation and a new method of starting computations have been
devised by a member of the Technical StaflF. There is room now only
for a bare outline of these methods, without going into the proof of
the formulas involved,

TRAJECTORY COMPUTATION.

The method of trajectory computation may best be described by
comparing it with that of Chapter VIII.

The first four or five lines are computed as laid down in that
chapter, or by the method hereinafter described. Thereafter, the
present method makes usei of ''anti-differences" of y" and x"
These are written A'^ and A*. Thus, if we have a tabulation of A"',
^"S y"j «> ^> Ci ®^« (<5f- P- 30), A* is the first difference of A~^;y'' is
the first differcAce of A"^ and hence the second difference of A"', etc

Now, since, for instance, 25 and 30, 93 and 98, 1000 and 1006,
all have the same difference, and hence either set could equally
well serve as A** to produce y" = 5, it becomes necessary to deter-
mine, in some manner, the initial values of A"* and A"'. By insert-
ing in formula 121 (post) the values of y', y", and the a, 6, and c of y"
for any given line, the value of A'* for that line can easily be
ascertained. Do this for two or three successive lines; and, if neces-
sary, adjust the values of A~^ thus foimd, so that the already tabu-
lated values of y" will be exactly the first differences of A"^

Similarly get the values of A"', from formula 122 (post) for several
successive lines, adjusting as before. Similarly get values of the
A-i and A' of x".

From this point on, proceed practically as in Chapter VIII, except
that we obtain «', as, y', and y, direcUy by the formulas of this sup-
plement,* instead of A«', A*, Ay', and Ay, as in Chapter VIH.
Integrate ahead by formulas 33 or 35 of that chapter. For the ten-
tative value of y, to use in getting log H, we may disregard all terms
of formula 34 except:

120 £y' dt^i (y\-lat)

Compute the tentative values of y'^ and x'' on the small sheet, as
in Chapter VIII, and enter them on the trajectory sheet. Obtain the
new A"S by algebraically adding the y" of its line to the A"* of the
line before. Obtain the new A"*, by algebraically adding the A"*
of its line to the A"* of the line before. Similarly for «".

1 A Technical Staff paper, accompanying a Sample Trajectory, gives additional formulas which
further lighten the labor of integration in special cases.

116

G. NEW METHODS OF TBAJBCTOKY COMPUTATION. 1^7

Then, instead of getting the increment of y' by formula 34 of
Chapter VIII, get y' itself hy:

1 111

121.' y't='i(V^-2^"*"i2^*""24 ^'""40^*^

and, instead of then integrating y' to get the increment of y, get y
iiself direct from y " by :

122. y, = i»(A-'-A- + ly"e-^)

Note that the first difference, a, does not occur in this formula;
and that i is squared. The first two terms in the parenth^es can, be
consolidated into A;;!;. The last term in the parentheses cj^n
iisoally be neglected or estimated.

x' can be found from formula 121 using x^' and its differences/ and
X (whiinever needed) from formula 122, in each forniida substituting:
«*s fpr all the j/'s.

No columns of differences of y', y, x\ or x need be carried on ^e
trajectory sheet, unless desired as a check on the smoothness with
which these functions are developing. Also, at least one less column
(rf differences need be carried for y" and x" than in Chapter VIIL
T]i^us, in place of the 25 colimms of figures tabulated on the trajec-
tojy ^eet in the method of that chapter, the present method cani
get along with 17 columns, as follows:

x, x', A ^ A-\ x"y a, 6, c, t, y, y', A^^ A^ y", a, 6, c.

The chief advantages of this new method are: It necessitates fewer
columns of figures; its integration formulas are more rapidly con-
vergen't than those of Chapter VIII; fourth differences, which are
apt to be very erratic in practice, are not employed in this method;
any errors in integrating by this method are noncumulative, except
of course as affecting x" and y" through the small-sheet computations^
whereas in the method of Chapter VIII errors have this effect and
in addition cumulate in their own columns. All that has to be paid
for this gain is a slight artificiality of method, and the slight additional
labor of starting the A"* and A" ^columns.

» « » ■ f ■ ■ ■

* If a and e are running so smoothly that at+\, C\+^ apd ct-i-^i can be estinaated,the iQlIowingfan be, sub:
ttituted for formula 121:

TUB farmula tias obvious advantages, U the time-interval is two seconds.

..i

118 COTJBSE m EICTEBIOB BALLISTICS.

STARTING THE TRAJECTORY.

The new method of starting trajectories is applicable either to the
foregoing method or to that of Chapter VIII, except on very jfliat
trajectories.

The values of the various elements for ^ = are computed as in
Chapter VIII. Tentative values for log Ex' and log Ey' tor t=i are
then extrapolated, by subtracting from the values for ^ = the
respective quantities DL Ex' and DL Ey', found by the formulas:

.^^ 10* DL Ex'^AE+By'

^^* 10* DLEy'^lO'DL Ex' +0

If i is other than one second, multiply each of these by i before
subtracting.

The factor 10* is included for convenience in connection with four
place logarithms. The decrements foimd by formulas 123 represent
the change in the fourth decimal place of the logarithm in question.
The logarithms mentioned in this chapter are all denary, although
natural logarithms were used in the derivation of the formulas.

Log^ and log B are given on a one-page table ("Auxiliaries for

Starting the Trajectory ") with the argument j?r^ • Cis from this table,

by means of the argument y'. These symbols should not be confused
with the three rotation coefficients of Chapter XIII, nor with the
ballistic coefficient C,
The following is the explanation of the symbols involved:

DL Ex' ^^\og Ex'
-4 = 10* Mn

If = log «« 0.434294 . . .

Only a rough interpolation for log A and log B is necessary. For
a y' less than 100, muUiply y' by 10, enter the table with the result
and mtdtiply by 10 the C thus obtained.

The tentative values of log Ex' and log Ey', resulting from the use
of the foregoing formulas, yield tentative values of x" and y", which
can be integrated in the manner of Chapter VIII, to y', y, and x\
with which to compute the column for t^i on the small sheet as in
that chapter.

G. NEW METHODS OF TRAJECTORY COMPUTATION. 119

Repeating the same process upon the final figures of the column
for <==z, we obtain tentative values of y', y, and x for t^2i. And so
^n, until enough figures are tabulated on the trajectory sheet to supply
sufficient differences for integration ahead. The use of this special
method can then be discontinued, and the trajectory completed,
either by the method of Chapter VIII or by the method discussed
at the b^inning of this supplement.

This special method of starting trajectories makes possible a very
close approximation of x" and y", and thus obviates the repeated
recomputation of the first few time intervab of the trajectory.'

I ■ ■■ ■ » III I ^ » »— *^^^-.^^— ^M^B^^ I ■ ■ ■ ^^^^^i— ^■^a^— ^^^^^^^ IIP! ■ ■ —^^1—— ^ ■! ■ ■ ■ ■■■^^^— ^ ■ ■»^^w^«i»..».». ■■. ■■■.— ■ .lai.M ■ ■!

* It is claimed that this method will enable the computer to begin a trajectory with oae«eoond interval!.

There is toaoB doabt as to the validity of this claim far very high veiodties or very low ballistic ooeflldents.

This method is to be used merely to yield approx i maie values of x" and f*, on which to base the usual

-method of successive approximations. It is mathematically equivalent to obtaining correct values for

C" and f" at <— !•, and assuming that this value remains unaltered during the interval. It therefoce

i^waterially reduces the number of approximations necessary to get a satistMitory valueof s' andr>

Supplement H.

NOTE ON ADVANaNG DIFFERENCE FORMULAS.

The student's attention is directed to the fact that, in most math-
ematical books that treat of interpolation and integration, advancing
differences are used almost exclusively, instead of the receding differ-
ences used throughout this book. .Therefore, the formulas of most
textbooks on finite differences will not be those of this book.

'Hie computations of ballistics that involve interpolation or inte-
gration are mostly of such a nature that the work continually lies
at or very near the last one of those values of the functicms already
derived and tabulated. (In the formula for "integration ahead,"
for instance, one limit of our integral, and, in fact, the entire integra-
tion interval, lies beyond the time of our latest estabUshed and tabu-
lated value of the function.) Hence, we are continually so situated
that the only differences available to us in this region are receding
differences. We are, therefore, unable to use the advancing differ-
ence formulas usually given in textbooks.

The student who wishes to construct advancing difference for-
mulas, for use in special cases, may use either of the methods given
in this note. It is, however, advised that all such formulas be plainly
labeled to indicate that they are to be used ordy with advancing
differences.

Advancing differences are formed thus:

That is, /a — /i is now called a^ (instead of aj) ; /, — 2/i +/o, which, being
the old aj — ^1 , was called 63, has now become (ii — ao = &o; ®^c*

It is to be noted that, whether re9eding or advancing differences
are used, the subtractions are always made with same direction; i. e.,
we always have/j— /i, a^ — a2j etc., never /1—/2 or ai — a^-

One method of obtaining any advancing difference formula from
the corresponding receding difference formula (see footnotes, pages
17 and 31) is as follows:

In the formula for receding differences, change the algebraic sign
of every odd-ordeved difference, the even-ordered remaining un-
changed. Also, change the sign of every t (and therefore of every
A^ throughout the equation.

The last condition requires that any integral shall, have the sign of
dt and also of both the limits changed. Thus the formula :

120

/d^= terms in/, a, 6, c, etc. (receding),
+1

H. NOTE ON ADVANCING DIFFERENCE FORMULAS, 121

will become

/•-a /•-a /•-!

+ Ifi — dt)—— \fdt=+ I /(ft = terms in/, a, &, c, etc. (advancing) ;

the coefficients of/ a, 6, c, etc., in one equation, being respectively
equal to the coefficients of/ a, 6, c, etc., in the other equation, but in
the second equation the coefficients of a, c, etc. (the odd-ordered differ-
ences), are each opposite in sign to the same coefficients in the other
equation. Referring to the schedule of page 30, it is noted that if
the first equation above uses/, a©, ho, Cq, etc., of the receding differ-
ence notation, then the second equation will use/, do, bo, Cq, etc., of
the advancing difference notation, which will be respectively equal to
/o, tti, 6a, C3, etc., of the receding differemce notation of the schedule.

The preceding process is reversible: i. e., exactly the same changes
are to be made in transforming any formula for advancing differences
into the corresponding formula for receding differences.

A full set of advancing difference formulas can be derived ab
initio, by tabulating/,/,/, etc., and the advancing differences;
and then performing the work of problems' 28 to 35, and 39 and 41;
noting that the algebraic signs of all subscripts, limits, and values
of ^ in general, in the statements of those problems, are to be changed.

The expression: ^'corresponding formula," which has been used in
the preceding discussion, needs some explanation. If a formula
Uses differences receding (or advancing) from /, to interpolate for
fny the *' corresponding'' formula in differences advancing (or receding)
from/ wiU interpolate f or/_ii. If a formula using differences receding

(or advancing) from/ gives the value of Ifdt, the ''corresponding"

formula in differences advancing (or receding) from/, will give — I fdt.

For example, the formula in receding differences, for ''integration
ahead" from/ transforms into a formula using advancing differences
for integrating over the interval back from /.

It should be noted that even in Chapter XIV, where the function
to be integrated is tabulated in reverse order of time, yet, since the
differences used recede with respect to the order of tabulation, only
receding difference formulas aretobe employed. (Seefootnote,p.30.)

INDEX.

A. Page.

A (rotation coefficient) 81,80

An 64 and note.

Acceleration 35

Deflection effect of increment in lateral . . 81

General effect of chaises in 61

AdTandng differences. See Differences.

Alger's charts 97

Ambiguity of partial derivative notation 10

Angle:

Complementary site 68

ofdepartore 35

cmrection formulas, see below.

effect of dianges in 60

computation (rf these effect 89

of fall 35

of inclination 35

of prpleetion 35

of site 36,71

. Angle of dei>artiire correction formulas:

effect of can t 71

of lump 70

of angle of site 71

in general. 72

. Anti-aircraft fire 109

Anti-differenoes 116

' Approximations, Successive. See Snoceseive.

.Amaud, J. J 8

Ascending branch 37

-Auxiliary variables:

Computation of. 83

Derivation of 53

Equations of 58, 83

firstintegral 56

Physical meaning of. ... ; 53, 50

B (function of velocity) 44^63

J9 (rotation coefDdent) 81,80

Ballistic Ck)efficient:

Comparison of French, Normal, and

Siaod 97

Dimensions of 44,107

Effect of nonstandard 61

Computation of this effect 80

Ballastic Tables:

American (new) 8

GAvre 97

Ingalls 8,41,97

Ballistic Wind 91

Range. 94

Cross 94

iBalUstics:

Exterior. See Exterior Ballistics.

Interior 62,97

The three parts of the science 38

Badiforth 39

Page.

Bennett, Prof. A. A 8

Blank forms. See Forms.

Bliss, Prof. G.A 8

Bracdalini 39

Brandi:

Ascending 37

Descending 37

C (ballistic coefficient) 38^97

Dimensions of 44,107

French, Normal, and Siaod C's. 97, 08

C (rotation coefficient) 81, 80

Cant:

effect on deflection 73

effect on departure * 71

Pivot of 1 73

Center of impact 36

Charts:

Alger's 97

Frendi 07

Coefficient:

Ballistic. See Ballistic Coefficient.

ofform 36,97

Rotation 81, 89

(Complementary site 6H

(computation:

by successive approximations 14

Forms* See Forms.

of differential cMrections 83

of trajectories—

by antidifference method 116

by rectangular method 47

Smoothingout 31,40

Starting. 118

tangent-reciprocal method 103

Computer's work, Three parts of. : 7

Computing sheet 47

Coordinate systems:

curved (the present standard) . . 7, 35^ 45, 68, 112

secant 45

tangent 45,67, 111

Correction formulas:

Angle of departure. See Angle.
Deflection. See Deflection.
Range. See Range.

Corrections, Differential. See Differential.

Curvature of the earth 67

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